Arts, Faculty of
Vancouver School of Economics
DSpace
UBCV
Corcuera Garcia, Paul Josemaria
Doctor of Philosophy - PhD
University of British Columbia
This dissertation consists of three distinct chapters in labor economics and applied econometrics. In the first chapter, I use detailed Peruvian social security records combined with firm-level financial data to analyze the impact of a minimum wage increase, enacted in Peru in 2016, on workers and firms, with a focus on the role of the informal sector in explaining these effects. I show that firms facing greater competition from the informal sector find it more difficult to adjust wages or prices in response, and are thus more likely to reduce employment. The second chapter presents a redistribution analysis of the minimum wage increase studied in Chapter 1. I show that low-wage formal employees' purchasing power increased, as the expected change in income is positive and exceeds the expected increase in expenses following the wage hike. I also show that most of the minimum wage pass-through to prices is borne by high-income individuals and households. However, redistribution towards low-income households is limited, as these households are largely composed of informal workers, who are unaffected by this policy. The third chapter proposes a method to estimate the joint cumulative distribution function of treatment effects across multiple outcomes in a difference-in-differences research design and demonstrates the validity of a multiplier bootstrap to construct uniform confidence bands.
Essays on Labor Economics andApplied EconometricsbyPaul Josemaria Corcuera GarciaB.Econ, Universidad de Piura, 2015M.A., The University of British Columbia, 2017A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE FACULTY OF GRADUATE AND POSTGRADUATE STUDIES(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)September 2024© Paul Josemaria Corcuera Garcia 2024The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Essays on Labor Economics and Applied Econometricssubmitted by Paul Josemar´ıa Corcuera Garc´ıa in partial fulfillment ofthe requirements for the degree of Doctor of Philosophyin EconomicsExamining Committee:Raffaele Saggio, Associate Professor, Economics, University of British ColumbiaSupervisorClaudio Ferraz, Professor, Economics, University of British ColumbiaUniversity ExaminerGuillermo Marshall, Associate Professor, Sauder School of Business, Uni-versity of British ColumbiaUniversity ExaminerFrancesco Amodio, Associate Professor, Economics, McGill UniversityExternal ExaminerAdditional Supervisory Committee Members:Thomas Lemieux, Professor, Economics, University of British ColumbiaSupervisory Committee MemberNicole Fortin, Professor, Economics, University of British ColumbiaSupervisory Committee MemberKyungchul Song, Professor, Economics, University of British ColumbiaSupervisory Committee MemberiiAbstractThis dissertation consists of three distinct chapters in labor economics andapplied econometrics. In the first chapter, I use detailed Peruvian socialsecurity records combined with firm-level financial data to analyze the im-pact of a minimum wage increase, enacted in Peru in 2016, on workers andfirms, with a focus on the role of the informal sector in explaining theseeffects. I show that firms facing greater competition from the informal sec-tor find it more difficult to adjust wages or prices in response, and are thusmore likely to reduce employment. The second chapter presents a redis-tribution analysis of the minimum wage increase studied in Chapter 1. Ishow that low-wage formal employees’ purchasing power increased, as theexpected change in income is positive and exceeds the expected increase inexpenses following the wage hike. I also show that most of the minimumwage pass-through to prices is borne by high-income individuals and house-holds. However, redistribution towards low-income households is limited, asthese households are largely composed of informal workers, who are unaf-fected by this policy. The third chapter proposes a method to estimate thejoint cumulative distribution function of treatment effects across multipleoutcomes in a difference-in-differences research design and demonstrates thevalidity of a multiplier bootstrap to construct uniform confidence bands.iiiLay SummaryMy dissertation consists of three distinct chapters in labor economics andapplied econometrics. The first chapter studies an increase of the minimumwage in Peru and its impact on workers and firms. I find that firms facinggreater competition from the informal sector find it more difficult to adjustwages or prices as a response to the minimum wage hike, which highlights thesalience of the informal sector in shaping the effects of the minimum wage indeveloping economies. On chapter 2, I perform a redistribution analysis ofthe minimum wage increase in Peru through changes in purchasing power.I show that low-wage formal workers’ purchasing power increased, and re-distribution towards low-income households was limited. The third chapterproposes a method to estimate the joint cumulative distribution functionof treatment effects across multiple outcomes in difference-in-differences re-search designs and demonstrates the validity of the multiplier bootstrap toperform inference.ivPrefaceChapters 1 and 2 are original, unpublished and independent work. Chapter 3is based on joint work with Professor Kyungchul Song. We both contributedto the study conception. I was responsible for deriving the theoretical resultsand preparing the draft manuscript shown in this dissertation.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 The 2016 Peruvian minimum wage increase and its impacton firms and workers . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Institutional Background . . . . . . . . . . . . . . . . . . . . 41.3 Conceptual Framework . . . . . . . . . . . . . . . . . . . . . 51.3.1 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3.2 Firm-level labor supply . . . . . . . . . . . . . . . . . 71.3.3 Firm’s problem . . . . . . . . . . . . . . . . . . . . . 91.3.4 Informal Sector . . . . . . . . . . . . . . . . . . . . . 91.3.5 Minimum Wage . . . . . . . . . . . . . . . . . . . . . 101.3.6 Incorporating imperfect competition in product mar-ket . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4.1 Social Security Data . . . . . . . . . . . . . . . . . . . 111.4.2 Annual Economic Survey . . . . . . . . . . . . . . . . 131.4.3 National Household Survey . . . . . . . . . . . . . . . 141.4.4 Summary Statistics . . . . . . . . . . . . . . . . . . . 14viTable of Contents1.5 Research Design: Firms . . . . . . . . . . . . . . . . . . . . . 151.6 Results: Firms . . . . . . . . . . . . . . . . . . . . . . . . . . 171.6.1 Employment and Wage Effects . . . . . . . . . . . . . 171.6.2 Other Firms’ Margins of Adjustment . . . . . . . . . 191.7 Research Design: Workers . . . . . . . . . . . . . . . . . . . 201.8 Results: Workers . . . . . . . . . . . . . . . . . . . . . . . . . 221.8.1 Wage, Employment and Reallocation Effects . . . . . 221.8.2 Do workers exit the formal sector? . . . . . . . . . . 241.9 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251.10 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.11 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 A redistribution analysis of the 2016 Peruvian minimumwage increase . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.2.1 Individual and Household Panel Data . . . . . . . . . 502.3 Research Design: Informal Workers . . . . . . . . . . . . . . 512.4 Research Design: Households . . . . . . . . . . . . . . . . . . 512.5 The Incidence of Minimum Wage in Peru . . . . . . . . . . . 522.5.1 Who pays for the minimum wage? . . . . . . . . . . . 522.5.2 Decomposing the change in labor cost . . . . . . . . . 532.5.3 Estimating the Change in Expenses . . . . . . . . . . 542.6 The Gains of the Minimum Wage in Peru . . . . . . . . . . . 542.7 A Redistribution Analysis of the Minimum Wage in Peru . . 552.7.1 Redistribution: Individual Analysis . . . . . . . . . . 552.7.2 Redistribution: Household Analysis . . . . . . . . . . 562.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.9 Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 582.10 Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633 Estimating the distribution of treatment effects in difference-in-differences: A deconvolution approach . . . . . . . . . . . 673.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 673.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.3 Estimation and Asymptotic Inference . . . . . . . . . . . . . 733.3.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . 733.3.2 Asymptotic Inference . . . . . . . . . . . . . . . . . . 743.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.4.1 Adding Covariates . . . . . . . . . . . . . . . . . . . . 79viiTable of Contents3.4.2 Staggered Adoption . . . . . . . . . . . . . . . . . . . 803.4.3 Beyond two outcomes . . . . . . . . . . . . . . . . . . 833.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.6.1 Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . 853.6.2 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . 913.6.3 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . 923.6.4 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . 1003.6.5 Proof of Corollary 1 . . . . . . . . . . . . . . . . . . . 105Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106AppendicesA Appendix to Chapter 1 . . . . . . . . . . . . . . . . . . . . . . 110A.1 Additional Figures . . . . . . . . . . . . . . . . . . . . . . . . 110A.2 Data Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . 121A.2.1 Linking PLAME and EEA . . . . . . . . . . . . . . . 121A.2.2 Firm concentration observed in EEA . . . . . . . . . 121A.3 Additional Empirical Evidence . . . . . . . . . . . . . . . . . 123A.4 Upper Tail Imputation . . . . . . . . . . . . . . . . . . . . . 124A.5 AKM Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 125B Appendix to Chapter 2 . . . . . . . . . . . . . . . . . . . . . . 126B.1 Additional Figures . . . . . . . . . . . . . . . . . . . . . . . . 126B.2 Alternative Computation of the Gains of the Minimum Wage 128viiiList of Tables1.1 Who are the minimum wage workers? . . . . . . . . . . . . . 391.2 Firm characteristics in PLAME dataset . . . . . . . . . . . . 401.3 Firm characteristics in EEA dataset . . . . . . . . . . . . . . 411.4 Who are the exposed firms? . . . . . . . . . . . . . . . . . . . 421.5 Employment and Wage Effects . . . . . . . . . . . . . . . . . 431.6 Effects of the Minimum Wage on Wages and Employment:Individual Approach . . . . . . . . . . . . . . . . . . . . . . . 441.7 Reallocation Effects of the Minimum Wage: Individual Ap-proach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451.8 Firms’ Margins of Adjustment (incl. closures) . . . . . . . . . 461.9 Firms’ Margins of Adjustment (excl. closures) . . . . . . . . . 471.10 Effect on Firm-Level Price Index in the Manufacturing Sector 482.1 Incidence of the Minimum Wage . . . . . . . . . . . . . . . . 642.2 Do workers benefit from MW increase? . . . . . . . . . . . . . 652.3 Do households benefit from MW increase? . . . . . . . . . . . 66ixList of Figures1.1 Minimum Wage to Median Earnings . . . . . . . . . . . . . . 271.2 Employment and Wage Effects: Firm Approach . . . . . . . . 281.3 Employment and Wage Effects, by Informality . . . . . . . . 291.4 Earnings Effects: Worker Approach . . . . . . . . . . . . . . . 301.5 Employment Effects: Worker Approach . . . . . . . . . . . . 311.6 Reallocation Effects: Worker Approach . . . . . . . . . . . . . 321.7 Employment and Wage Effects, by Informality . . . . . . . . 331.8 Frequency of earnings distribution in 2016 and 2018 . . . . . 341.9 Firms’ Margins of Adjustment . . . . . . . . . . . . . . . . . 351.10 Effect on Laspeyres Price Index in Manufacture . . . . . . . . 361.11 Effects on Labor Cost and Revenue, by Informality . . . . . . 371.12 Aggregate Employment Change in the Formal Sector . . . . . 382.1 Consumption of goods produced using MW workers . . . . . 582.2 Expected gains from MW . . . . . . . . . . . . . . . . . . . . 592.3 Effects on Household Income and Consumption . . . . . . . . 602.4 Remaining in Informal Sector . . . . . . . . . . . . . . . . . . 612.5 Effects on Income for Informal Incumbents . . . . . . . . . . 62A.1 Macroeconomic Variables between 2011 and 2017 . . . . . . . 110A.2 Employment Effects, Extensive and Intensive Margins . . . . 111A.3 Employment Effects, by Sector: Firm Approach . . . . . . . . 112A.4 Composition of Labor Force in Peru . . . . . . . . . . . . . . 113A.5 Number of Firms in EEA across provinces . . . . . . . . . . . 114A.6 Linearity between Exposure and Wage Growth . . . . . . . . 115A.7 Linearity between Exposure and Employment Growth . . . . 115A.8 Own-Wage Elasticity in the Literature . . . . . . . . . . . . . 116A.9 Out-of-sample prediction of Fraction Affected (FA) . . . . . . 117A.10 Effects on Type of Contract: Worker Approach . . . . . . . . 118A.11 Effects on J2J Transition: Worker Approach . . . . . . . . . . 119A.12 Effects on Firm Size: Worker Approach . . . . . . . . . . . . 120xList of FiguresB.1 Effect on Household’s Consumption Price Index . . . . . . . . 126B.2 Expected gains from MW . . . . . . . . . . . . . . . . . . . . 127xiAcknowledgementsI am extremely grateful to the members of my supervisory committee fortheir invaluable advice and support. I am especially thankful to RaffaeleSaggio for his continuous encouragement and support. I particularly thankhim for believing in this project when it was still at a very early stage. Iam proud to have been among the first PhD students under his guidance.He is truly an inspiring example of a great scholar and, just as importantly,a kind person. I am also profoundly grateful to Kyungchul Song for hisvaluable conversations about econometrics, the PhD journey, and other lifelessons. I want to express my heartfelt appreciation to Nicole Fortin andThomas Lemieux for their advice and for being incredible role models, bothacademically and personally.I also benefited from conversations with many other UBC faculty mem-bers, who provided me with great support. First, I am grateful to VadimMarmer, from whom I learned tremendously as his teaching assistant formany years. I will be forever inspired by his teaching approach and hisdeep care for his students. I would also like to thank David Green for verythoughtful discussions and for helping me request the dataset used in thisdissertation. Similarly, I would also like to thank Claudio Ferraz for mak-ing me, as well as every other student from Latin America, feel a little bitcloser to home. Finally, I want to thank all the members of the econometricsreading group, from whom I learned valuable lessons about teamwork andcollaboration when it comes to learning new and difficult topics.I would also like to thank the Peruvian Ministry of Labor for grantingpermission to use a de-identified version of their matched employer-employeedataset; I am particularly indebted to David Tenorio, Katia Samanamud,and Viviana Cruzado for their help with the data request.I’ve been incredibly fortunate to share my path through graduate schooland Vancouver city with many great people. First, I would like to thankevery single member of my graduate cohort: you are all truly one of akind. I am also particularly grateful to Angel Espinoza, Fernanda Hurtado,Francisco Eslava, Pablo Valenzuela, Catalina Bobadilla, Pascuel Plotkin,Valentina Rutigliano, Daniel Dulanto, Luis A´lvaro A´lvarez, Sebastian Melo,xiiAcknowledgementsXiomara Pulido, Bernardo Ribeiro, Adrian Herra´n, Amirhossein Tavakoli,Fernando Secco, Federico Ricca, Giulia Lo Forte, Bruno Esposito, RatzanyelRinco´n, Ieda Matavelli, Deivis Angeli, Felipe Sepu´lveda, Bernardita Sepu´lveda,Tiago Bonomo, Daniel Guzma´n, Alfonso Herna´ndez, Carla Srebot, Sebastia´nGo´mez, Valentina Vargas, Spreeha Aggarwal, Prerna Kundu, Alejandro Ro-jas, Gian Pierre Trelles, Diana Ricciulli, Andre´s Reyes, Keith Pe´rez, andErick Lino for their friendship and support. I want to thank Samuel Gyet-vay, Catherine van der List, Yige Duan, Jan Rosa, Pierre-Loup Beauregard,and all other present and past members of the UBC Labour Group. I wouldalso like to especially thank Germa´n Vega and Claudy Dedios for welcomingus into their home and making us feel like family during the first few years ofmy PhD journey. My family is forever indebted to Daniel Jaramillo, DanielaChiriboga, D’angela Co´rdova, Dar´ıo Loyola, Dashleen Kaur, Ana ClaudiaPalacios and Enrique Chueca for their invaluable help during a very chaoticfirst week postpartum and for their invaluable friendship.I am very grateful to my wonderful friends and previous mentors backin Peru. I thank Marcos Agurto and Gabriel Natividad for introducing meto this journey quite a few years ago and for their guidance. Thank youto my amazing friends for keeping me sane during the many lows of mygraduate studies through our Whatsapp groups. I apologize for missing outon many wonderful moments with all of you and for not being there duringyour difficult times. This thesis would not have been possible without theoccasional Quarter Pounder, diet coke, and beer.My deepest gratitude goes to my parents, Paul and Mariela, and mybrothers, Juan Pablo and Alvaro, for their unconditional love and support.I also thank my in-laws, Oscar, Victoria, and Oscar Jr., for always checkingin on us and visiting to help before and after Olivia was born. This is alsodedicated to Gabriel and Laura, and to the loving memory of my grandma,Celia, to whom I couldn’t say goodbye. I am also thankful to my amazingLuna, the greatest pet companion anyone could ask for.Lastly, and most importantly, I want to thank my dear wife: Claudia.We have shared both tough and good times together over the past 11 years.Gracias por creer en mı´ y ser mi fortaleza. Este, y todos mis dema´s logros,te pertenecen en la misma medida. Te amo.xiiiDedicationA Olivia, per aspera ad astra.xivChapter 1The 2016 Peruvian minimumwage increase and its impacton firms and workers1.1 IntroductionAre the effects of minimum wage policies in developing countries qualita-tively different due to the presence of sizable informal self-employment? Inthis chapter, I provide a comprehensive assessment of the impacts of theminimum wage by exploiting an increase in the minimum wage in Peru, andstudy the salience of the informal self-employment to explain the hetero-geneity of these effects. The main results of this chapter set the stage forthe evaluation in Chapter 2 of changes in purchasing power across differenttypes of workers and households.In this chapter, I introduce the institutional background around the min-imum wage increase enacted in Peru in 2016. I then explain the main datasources used in the first two chapters of this dissertation. Next, I introducea simple extension of the model in Haanwinckel (2023) that can explain howthe effects of the minimum wage depend on the employment size of the infor-mal sector. This chapter concludes by presenting empirical evidence of theimpacts of this policy on firms and workers, and confirming the predictionin the model regarding the heterogeneity based on the presence of informalself-employment.Economic theory suggests that minimum wage increases can be beneficialthrough two main channels: the reduction of monopsony power (efficiencychannel), and shifting resources from business owners to lower-income in-dividuals and households (redistribution channel). Many low- and middle-income countries are characterized by frequent transitions between informalself-employment and low-wage work (Donovan et al., 2023), where the formertypically exists as a form of subsistence employment for many individuals(Breza et al., 2021). In these economies, we also typically observe com-11.1. Introductionpetition between the informal and formal sector (Ulyssea, 2018; Hsieh andKlenow, 2009). If the informal sector competes with formal firms in the la-bor and product market, then the effects of minimum wage policies, as wellas their potential benefits, can be a priori unclear. Whether this featureof developing countries matters to understand the effects of minimum wagepolicies is, ultimately, an empirical question.In this chapter I provide novel empirical evidence that the large presenceof informal self-employment shapes market power and it is thus key to under-stand why the minimum wage can generate negative employment effects forsome workers (Amodio et al., 2023; Felix, 2022). I formalize the key mech-anism with a theoretical framework that incorporates both selection intoself-employment and involuntary self-employment, as well as oligopsonisticcompetition among firms (Berger et al., 2022).I begin by laying out a model where workers choose employment basedon a nested discrete choice among wage employment and the firm sector,and then within firms in the firm sector. In addition to this, firms competein an oligopsonistic style and are able to endogenously choose a cutoff ofworker productivity from which they will only hire (Berger, Herkenhoff, andMongey, 2024; Haanwinckel, 2023). The presence of large informal self-employment changes the structure of both labor and product markets in thefollowing ways. First, low-wage workers have an easily accessible outsideoption that makes firms post wages closer to a competitive level. Next,due to imperfect competition in the product market between self-employedand firms, the product prices will be pushed closer to competitive levelstoo. This leads to the prediction that minimum wage increases lead firmsexposed to more competition with informal self-employed units to find itharder to adjust wages and prices in response. Therefore, in such cases,one can expect to observe more layoffs, closures and reallocation towardsinformal self-employment.To perform the empirical approach in this paper, I exploit a nationalminimum wage hike in Peru that was enacted in May, 2016. First, usinga firm-level design I find that the minimum wage increase had a negativeand sizable employment effect, larger than recent estimates in the contextof developed economies (Cengiz et al., 2019, 2022; Harasztosi and Lindner,2019). In particular, my estimates imply a short-run own-wage elasticity ofapproximately -0.85 (SE 0.06) one year after the minimum wage increase,and -1.12 (SE 0.08) two years after. By linking occupation-level informal-ity measures constructed from a household survey to a matched employer-employee dataset, I construct firm-level measures of exposure to the informalsector. My estimates show that firms that are highly exposed to informality21.1. Introductionhave muted effects on wages and revenues, and larger employment losses.Next, using a worker-level design I show that not all workers were able to re-allocate towards higher quality firms and workers in occupations with moreoverlap with the informal sector were less likely to remain in the formal sec-tor. I conclude the analysis by combining on the research design developedin Giupponi et al. (2024) with detailed survey data, and show that the sizeof the formal sector decreased after the minimum wage hike.The findings in this chapter indicate that in labor markets with largeinformal self-employment, the wage markdowns are smaller due to a morecompetitive landscape, thus hindering the potential of minimum wages toreduce monopsony power. Additionally, at the expense of some workersreallocating towards the informal sector, the low wage formal workers thatremain formally employed after the minimum wage hike tend to receive awage boost up to the 80th percentile of the earnings distribution1. Theseresults set the stage for the upcoming chapter, in which I seek to answerwhether there is any potential benefit of the minimum wage through theredistribution channel, by studying changes in purchasing power.Related Literature This chapter contributes to several strands of theliterature. First, I contribute to the empirical literature that examines theeffect of the minimum wage on labor market outcomes (Card and Krueger,1994; Dube et al., 2016; Cengiz et al., 2019). In particular, the contextstudied in this paper corresponds to a country where the bite of the mini-mum wage is large and thus may yield qualitatively different responses thansmaller minimum wage hikes. The empirical approach in this paper relieson a comprehensive analysis of the impacts of the minimum wage based onfirm-level exposure (Harasztosi and Lindner, 2019) complemented with aworker-level research design as in Dustmann et al. (2022).Next, this chapter contributes to the literature on market power andhow they operate in developing countries. Evidence in the U.S. shows thatemployers exert market power (Berger et al., 2022; Azar et al., 2022). Trans-lating the micro-foundations towards developing countries requires incorpo-rating how the informal sector interacts with respect to workers and firms.Most notably, Amodio, Medina, and Morlacco (2023) propose a frameworkwhere sorting between wage work and self-employment is based on Roy’s(1951) selection model. A limitation of this framework in the context ofminimum wages is that it does not incorporate involuntary reallocation to-1This is consistent with recent evidence of the distributional gains of the minimumwage in Brasil (Engbom and Moser, 2022).31.2. Institutional Backgroundwards the outside option sector (that is, the existence of excess supply whenthe minimum wage is large enough). Therefore, in this paper I adapt thetheoretical foundation in Haanwinckel (2023) that describes how firms canendogenously choose a cutoff of worker productivity from which they do notwant to hire from, and captures the same idea as rationing labor (Bergeret al., 2022). I incorporate oligopsonistic competition that will generate agradient of market power and the presence of the informal sector.In addition, this paper relates to a vast literature related to informalityand labor markets (Meghir et al., 2019; Jales, 2019; Ulyssea, 2018, 2020).In particular, this paper sheds light on which theories of informality (e.g.,Haanwinckel, 2021) can be ruled out based on quasi-experimental evidencein low- and middle-income countries that are similar to the Peruvian context.1.2 Institutional BackgroundIn Peru, most workers are informal and self-employed. Informal firms aredefined as those firms that are not registered under tax authorities. Inaddition, a worker is classified as having informal labor if she reports (i) nothaving health insurance, or (ii) is self-employed, is classified as working foran informal firm, and has five or fewer employees. Figure A.4 shows that theset of informal firms, henceforth referred as informal sector, in much largerin terms of employment relative to the set of formal firms. Furthermore,more than half of the individuals in the informal sector can be classifiedas informal self-employed in my data. This can be contrasted with a verysmall unemployment rate that sits around 3%. Due to the large prominenceof self-employment within the informal sector, I will use the terms informalsector and informal self-employment interchangeably throughout the rest ofthis dissertation 2.The minimum wage is set in terms of monthly earnings of its nationalcurrency, soles (henceforth referred as PEN ), and these are set at a nationallevel only3. Typically, there is loose enforcement of some labor contract con-ditions such as the number of hours worked. Therefore, I will use the termswages and earnings interchangeably throughout the rest of this dissertation.As opposed to other countries in the region (e.g., Ecuador, Colombia, Chile),2Additionally, close to 90% of self-employed individuals in my data are classified asinformal. Therefore, whenever I mention self-employment I will implicitly refer to informalself-employment.3In Peru it is called Remuneracio´n Minima Vital (RMV). This quantity correspond tofull-time employment of 8 hours per day, or, equivalently, 48 hours per week. Workers inthe mining industry have a 25% larger RMV.41.3. Conceptual Frameworkthe minimum wage increases have no particular periodicity, and often de-pend on the political interest of the president at that particular period.On March, 2016, the president announced that the national minimumwage would increase from 750 PEN to 850 PEN, which corresponds to anapproximately 13 percent increase in nominal terms, and 15 percent in realterms. This change came into force on May 1st, 2016. Figure 1.1 shows thatthe ratio of the minimum wage with respect to the median earnings (amongformally employed workers) in Peru sits around 0.6. Therefore, the bite ofthe minimum wage in Peru was particularly large around the time of thepolicy, and had the potential to destroy a significant number of jobs.The Peruvian economy was characterized by robust economic growthin the years surrounding the implementation of the minimum wage policy.Over the period of 2011 to 2017, nominal GDP grew by 44% (see Panel (a)of Figure A.1). At the same time, unemployment was stable around 3.2%throughout the 2011-2016 period (see Panel (b)). The 2014-2016 period wascharacterized by some slight increase in inflation (Panel (c)), which laterreverted back towards values within 1% and 3%. These time series showthat the Peruvian economy was stable throughout these years, which willpose an important feature for part of the empirical strategy in this chapter.1.3 Conceptual FrameworkIn this section, I outline a partial equilibrium model of oligopsonistic com-petition for labor that incorporates transitions from wage employment andself-employment that can be involuntary due to the existence of the mini-mum wage. The purpose of this model is to illustrate the key role of theinformal sector and how it shapes market power, which will then help in-terpret the empirical results of this paper. In particular, I show that thelabor supply elasticity depends on the employment concentration of the in-formal sector, and thus affect the wage markdowns and employment choicesof formal firms.On the labor supply side, workers maximize indirect utility and sortbased on a nested discrete choice. These workers first decide between be-tween self-employment and wage employment, and then among firms withinthe wage sector. In the labor demand side there is a minimum wage andfirms can choose which workers to hire from based on their productivity4. I4This concept is similar to what Berger et al. (2022) refer to as rationing. It capturesthe idea that firms can endogenously choose to “close its doors” to some workers and thusgenerate excess supply when there is a high minimum wage.51.3. Conceptual Frameworkuse this to rationalize the existence of both voluntary and involuntary tran-sitions towards self-employment. I then incorporate imperfect competitionin the product market, in which the mechanism is qualitatively similar tothe one in the labor market.I use this model to interpret my empirical analysis as suggesting that,when there is a high degree of competition between firms and the informalsector for labor and product, a minimum wage hike is more likely to induceinvoluntary transitions towards self-employment. This is due to firms payingwages closer to competitive level and to a reduced capability to pass-throughlabor cost to price increases.1.3.1 SetupLabor markets are built upon the model of monopsonistic competition amongfirms described in Haanwinckel (2023). As in Card et al. (2018), the modelin Haanwinckel (2023) assumes that firms do not interact strategically. Iwill relax this assumption to allow for oligopsonistic competition. In suchenvironment, the size - in terms of employment - of the self-employmentsector will be relevant to determine the labor supply elasticity to the firm.Workers are ex-ante heterogeneous in their efficiency units ε ∼ G(·), thatis supported over the real line. This means that there will be a distinctionbetween quantity of workers n, and quantities of labor ℓ. I also denoteworker earnings as y and the price for labor w.As in Haanwinckel (2023), firms have to pay at least a total compensationof y, which I will refer to as the minimum wage5. Because some workerswith low ε can have a marginal product of labor smaller than y, firms j ∈{1, . . . , J} are allowed to reject workers with efficiency units smaller thanan endogenously choice εj . Therefore, this cutoff which will also depend ontheir productivity zj ∼ F (·), so that higher productivity firms can affordhiring workers with smaller efficiency units.Formally, the household preferences are given byU(c, j) = c · [exp(ηi,j)]1λ ,where c is a final good composed of firm-produced goods and the goodproduced by the self-employment sector. In particular, I assume these are5In this model, there is no variation in hours worked.61.3. Conceptual Frameworkcombined using a CES aggregatorqF = J∑j=1γjqσ−1σj σσ−1 , c = [αF q ρρ−1F + αSq ρρ−1S ] ρρ−1 ,where qF denotes firm production and qS denotes production at the infor-mal sector. For the time being, product markets will be assumed to becompetitive.Furthermore, aggregate prices are given bypF = J∑j=1γσp1−σj 11−σ , P = [αρF p1−ρF + αρSp1−ρS ] 11−ρ1.3.2 Firm-level labor supplyThe timing of the labor market is as follows. Firms post rejection cutoffs εjand wages wj . The self-employment sector operates as one large firm thatpost a wage wS . Second, workers observe firm choices in the wage sector(εj , wj), as well as wS . Based on these, they choose employment option thatmaximizes indirect utility. Third, firms observe ε of workers who appliedand hire those with ε > εj . Finally, production occurs and workers are paid.Rejected workers become unemployed and earn zero income.The choices in the second step will be based on a nested discrete choice.In particular, let j = 0 denote self-employment option and j = 1, . . . , Jdenote an employment option at one of the firms. Workers draw a vectorof idiosyncratic taste shifters {ηi,j}Jj=0. These preference shocks have acumulative distribution function given byCDF({ηi,j}Jj=0) = exp{− exp(ηi,0)− J∑j=1exp(−ηi,j · βλ)βλ},where the parameter β governs the correlation of taste shocks within the firmsector, and the parameter λ governs the correlation across self-employmentand the firm sector. I will assume that β ≥ λ.The indirect utility at different employment options are given byVi(ε, j) = exp (λ log (εwS) + ηi,S)1λ if j = 0,Vi(ε, j) = 1{ε ≥ εj} exp(λ log(max{εwj , y})+ ηi,j) 1λ if j ≥ 1.71.3. Conceptual FrameworkIn addition, the probabilities are given byPrinformal(ε,w) =(εwS)λ(εwS)λ +Ωλε,Prformal, j(ε,w) =Ωλε(εwS)λ +Ωλε1{ε ≥ εj}max{y, εwj}βΩβε,where Ωε =(∑Jk=1 1{ε ≥ εk}max{y, εwk}β)1/βis a measure for demandof skills among firms. High values of Ωε mean that many firms are postinghigh wages and willing to hire workers with ε efficiency units, despite theminimum wage. Importantly, I will allow firms to internalize that theirchoice of wages and cutoffs affects poaching from other employment options.The number of workers choosing firm j and the resulting supply of laborin efficiency units are given byn(wj , εj , w−j) = N∫ ∞εjΩλε(εwS)λ +Ωλεmax{y, εwj}βΩβεdG(ε),ℓ(wj , εj , w−j) = N∫ ∞εjΩλε(εwS)λ +Ωλεmax{y, εwj}βΩβεεdG(ε).Next, labor costs are given byC(wj , εj , w−j) = N∫ ∞εjΩλε(εwS)λ +Ωλεmax{y, εwj}β+1ΩβεdG(ε)The elasticity of labor supply, denoted as eSj , is given byeSj = β− (β − λ)∞∫εjΩλε(εwS)λ +Ωλε(εwj)2βΩ2βεε dG(ε) ∞∫εjΩλε(εwS)λ +Ωλε(εwj)βΩβεε dG(ε)−1︸ ︷︷ ︸Concentration within firm sector− λ∞∫εjΩ2λε[(εwS)λ +Ωλε ]2(εwj)2βΩ2βεε dG(ε) ∞∫εjΩλε(εwS)λ +Ωλε(εwj)βΩβεε dG(ε)−1︸ ︷︷ ︸Concentration overallThis differs the standard monopsony model with discrete choice (Cardet al., 2018), in which firms ignore their contribution to Ωε due to beingsmall relative to the size of the labor market. In that model, the elasticity81.3. Conceptual Frameworkof labor supply will be given by β. On the other hand, we will obtain suchresult in my proposed model only if the firm is very small relative to thelabor market. Furthermore, as firms become more concentrated the laborsupply elasticity tends to zero.1.3.3 Firm’s problemFirms choose the posted wages per efficiency unit and the productivity cutoffthat maximizes profitsmaxwj ,εjpjzjf(ℓ(wj , εj , w−j))− C(wj , εj , w−j),and the first order conditions yieldpjzjf′(ℓ) =(1 +1eSj)wj (1.1)pjzjf′(ℓ)εj = y (1.2)The first condition states that wages are set as a markdown of themarginal revenue product of labor, and this markdown depends on the firms’employment concentration. The second condition states that the thresholdis selected optimally such that the lowest productivity worker is compen-sated for their labor with the minimum wage.This result implies that firms that are more concentrated will be ableto markdown wages more so than firms with less concentration. The lessproductive firms will have a low marginal revenue product and will be smallrelative to the market, thus will be more likely to reduce employment as aresponse to the minimum wage increase.1.3.4 Informal SectorProduction at the informal sector is linear en efficiency units of labor:qS = ℓS , ℓS = N∫ ∞εj(εwS)λ(εwS)λ +ΩλεεdG(ε).That is, we can think of this sector as a very large firm that sells aproduct at marginal cost pS =WS .91.3. Conceptual Framework1.3.5 Minimum WageIn this section I will explain how minimum wages operate in this framework.In this model where firm concentration matters, each firm internalizes thatincreasing their posted wage wj will attract workers who are supplying laborto other firms, or choosing to be self-employed. Thus low-productivity firmswill be much less concentrated than high-productivity firms. Furthermore,high-productivity firms are able to afford hiring workers with lower ε thanthe other type of firms.An increase in the minimum wage increase the hiring thresholds of allfirms εj , as can be seen from Equation (1.2). Furthermore, as stated in Equa-tion (1.1) firms with low zj will post lower wages due to a lower marginalrevenue product. In other words, low-quality firms will pay wages closer tocompetitive levels, will be more atomistic, and will face a larger bite with re-spect to the minimum wage. Under the assumption of a concave productionfunction f(·), firms with lowest productivity in the market will have to setlarger hiring thresholds compared to a monopsonistic competition scenario.This mechanism is consistent with the theoretical model proposed inBerger et al. (2022), in which oligopsonistic competition reduces the po-tential efficiency gains from the minimum wage. In Berger et al. (2022) aminimum wage increase labor market power of surviving firms due to real-location. In this model, however, labor market power can be reduced whenlabor is reallocated out of the formal sector and into self-employment.1.3.6 Incorporating imperfect competition in productmarketThe discussion so far shows that when the size of the informal sector islarge, de-facto labor market power is reduced due to smaller employmentconcentration of firms. This implies that it will be harder for some firms toadjust wages when there is an increase in the minimum wage. In addition tothis mechanism, another response from firms is to pass-through the increasedlabor costs to prices (Harasztosi and Lindner, 2019).The mechanism in this case will be similar to the case of the labor market:oligopolistic competition and concentration restricts firms to adjust prices inresponse to a minimum wage hike. In particular, by allowing formal firms tointernalize that their pricing decision affects aggregate prices pF , we obtain101.4. Datathe modified first order conditions:pj(1− 1eQj)zjf′(ℓ) =(1 +1eSj)wj (1.3)pj(1− 1eQj)zjf′(ℓ)εj = y, (1.4)where eQj ≡ ∂ log qj∂ log pj denotes the product demand elasticity. Equations(1.3) and (1.4) show that under imperfect competition in the product mar-ket, firms have more freedom to adjust in response to a minimum wageincrease. For example, a firm that has a large degree of concentration in theproduct market can increase the price of the good without having to layoffany worker.1.4 DataThis section describes the data used for the empirical analysis. Section 1.4.1describes the administrative records that represent my primary data source.Section 1.4.2 outlines the firm-level financial dataset that complements thefirm-level analysis. Section 1.4.3 describes the household survey used toobtain measures of informality status among workers. Summary statisticsare presented in Section 1.4.4. Appendix A.2 provides further details aboutthese data sources.1.4.1 Social Security DataMy main data source comes from social security records collected monthlyby the Peruvian National Superintendency of Customs and Tax Adminis-tration (Superintendencia Nacional de Aduanas y de Administracio´n Trib-utaria, SUNAT), and stored by the Peruvian Ministry of Labor (Ministeriode Trabajo y Promocio´n de Empleo, MTPE). This dataset provides informa-tion, between January-2015 and December-2019, on the universe of formalemployees working at formal firms. I define formal firms as those that areregistered under the tax authority, SUNAT. I use the information recordedas of February in each year to perform the empirical analysis.This administrative data - henceforth labelled as PLAME - is a matchedemployer-employee dataset that contains information on approximately 3million workers and their current jobs. For each worker-job combination oc-curred in a given year, I have information on earnings, six-digit occupation111.4. Datacodes, industry, labor contract information, as well as detailed demographicinformation on the employee. 6 Individuals who participate in the informalsector are not observed in this dataset, which means that we cannot distin-guish unemployment, non-participation in the labor force, and participationin the informal sector from these records.A feature of this dataset that is instrumental for my purposes is thatit provides detailed information about each workers’ occupation. In myanalysis I leverage measures of informality at the occupation level that I canmerge to the occupation codes in this dataset. Furthermore, there is richinformation on the type of contract, which allows me to observe whether aworker is hired under a temporary vs permanent contract.My analysis involves two main empirical approaches. The first empiricalapproach is at the firm level and compares the trajectories of firms who werehighly exposed to the minimum wage versus firms who were not exposed tothe policy. To implement this approach, I collapse the information at thefirm-year level and exclude some sectors. I omit the finance sector, commu-nitary services, household domestic services, non-governmental, and firmswith missing industry information. I focus on firms that existed between2015 and 20167, and I drop firms with no wage cost information in eitherof those years. I also drop firms with top 1 percent and bottom 1 percentemployment growth rate between pre-policy years, as well as firms wherethe average wage per worker was less than 90 percent of the minimum wagein any year between 2015 and 2016. In addition, I omit firms with less than5 employees on average between 2015 and 2016 from my analysis.8 The finaldataset includes 23,477 firms that represent around 1.3 million workers (orclose to a third of the total formal employment in Peru).The second empirical approach compares job trajectories of workers whoearned less or close to the minimum wage prior to its increase with the jobtrajectories of workers who earned a substantially higher wage, in the samespirit as Dustmann et al. (2022). To do so, I keep the highest paying jobfor every worker-year combination and those who earned a monthly earningbetween 550 and 2250 PEN - as of February 29th - in 2016. I follow theseindividuals from 2015 to 2017. I focus on workers who were between 18 and6Earnings do not include overtime payment or bonuses. Earnings are top-coded at the95th percentile of each year (around 7,500 PEN in terms of monthly earnings).7Since I use the records as of February in each year, these two years correspond to thepre-policy periods.8The reason for this is that the firm-level financial information mostly covers medium-to-large firms. By imposing this restriction, it is easier to compare and reconcile theanalysis across different data sources.121.4. Data60 years old throughout the period of analysis and omit workers who werehired under part-time contracts.91.4.2 Annual Economic SurveyI complement the matched employer-employee data with a firm-level cross-sectional dataset. This dataset is effectively a census for medium and largefirms —based on their annual net revenues— and provides survey-qualityinformation for smaller firms that is representative at a regional and indus-try level. This dataset - henceforth referred to as EEA - include informa-tion at the annual level on employment counts across different categories(e.g., across gender, managerial status, permanent versus temporary con-tracts), geographical location, balance sheet information, and it also recordsproduct-level information such as quantities and prices for a subset of firms.Unfortunately, both PLAME and this dataset are de-identified and thustheir firm identifiers cannot be matched across datasets. To circumvent thisproblem, I follow an approach described in Harasztosi and Lindner (2019),Appendix A5. In particular, I use a battery of observable characteristicsat the firm level such as employment counts and average wage cost perworker across different occupational categories and gender status to predictthe fraction affected at the firms in the EEA data using machine learningmethods. The preferred model corresponds to a random (regression) forest.The model performs well in terms of out-of-sample prediction: the slope ofthe regression between actual and predicted fraction affected is 0.98, whichis very close to one. Further discussion of this procedure can be found inAppendix A.2. As a result, I am able to apply the same firm-level empiricalstrategy on outcomes found either in PLAME or EEA, and can provide acomprehensive analysis on the firms’ margins of response to the minimumwage increase.This dataset records information as of the end of December in each year.Therefore, for the empirical analysis I restrict to firms that existed between2014 and 2015 and had at least 1 employee in those years. I exclude firmsthat registered employment or wage cost equal to zero in either 2014 and2015. Additionally, I drop firms with top 1 percent and bottom 1 percentemployment and revenue growth rate between pre-policy years, as well asfirms where the average wage per worker was less than 90 percent of theminimum wage in any year between 2014 and 2015. Finally, I omit firms9I do not observe hours in my main data source. This restriction helps me focus onworkers who are more likely working full-time.131.4. Datawith less than 5 employees on average between 2014 and 2015 from myanalysis. The final dataset contains 4,343 firms observed in 2014 and 2015.1.4.3 National Household SurveyFinally, my third data source is the Peruvian National Household Survey(ENAHO), collected by the Peruvian National Institute of Statistics andInformation (Instituto Nacional de Estad´ısticas e Informa´tica, INEI). Thisdataset is cross-sectional but also follows some households over time as arotating panel. Among many other characteristics of the household mem-bers, this dataset gathers information on wages, employment characteristicsand household expenses. Starting from 2014, INEI also computes a set ofvariables that measure the informality status of workers and their firms 10I use the occupation codes in this dataset to collapse informality mea-sures at this level and match with the employer-employee data. For individual-level analysis using this dataset, I restrict the sample to workers who workbetween 35 and 72 hours per week and who were between 18 and 60 years oldwithin the period of analysis.11 One problem with the information on hoursin this data is that some individuals report actual versus contractual hours.This sample restriction attempts to mitigate this problem by assuming thatworkers who report to work more than 35 hours are full-time workers.1.4.4 Summary StatisticsIn this section I provide some summary statistics from the main datasetsused in the empirical results of the first two chapters of this dissertation.First, I provide some information about the workers and firms’ characteris-tics in the PLAME dataset. And next, I summarize some firm characteristicsin the EEA dataset.Table 1.1 provides a description of the characteristics of workers thatearned below the new minimum wage (850 PEN ) in early 2016 (whom I’llrefer as minimum wage workers), workers who earned slightly above thenew minimum wage, and then workers who are higher up in the earningsdistribution. Minimum wage workers are, compared to workers who earnmore than 1250 PEN per month, more likely to be residing outside of Lima,female, low-skilled, and younger than 24 years old; they are also more likely10These measures are inferred from key questions in the survey, and consistent withinformality definitions used by the International Labour Organization, ILO.11The exception to this rule are workers who reported to have worked a smaller numberof hours due to illness or vacations and receive a monthly salary.141.5. Research Design: Firmsto hold a permanent contract. Furthermore, minimum wage workers aremore likely to be working in small firms and in the non-tradable sector.Next, Table 1.2 reports some characteristics at the firm-level in 2016,aggregating all sectors and then for the three main sectors in terms of em-ployment size: manufacture, commerce, and services. The average firm inmy sample employs 115 workers, has 13 years of existence, and a femaleshare of approximately 33% of its workforce. The median fraction of work-ers affected by the minimum wage hike for the average firm is 17 percent.As expected, firms in the services sector employ less workers on average, andare also more exposed to the minimum wage than those in tradable sectors.Table 1.3 shows the characteristics of firms in the EEA dataset, whichcontains financial information. I compute statistics the year prior to theminimum wage hike up to the year after the increase. Firms in this sam-ple are, by construction, slightly older and with more employees than myPLAME estimation sample. In particular, in this dataset the average firmage in 2016 is 17.69, whereas it is 13.22 in the matched employer-employeedataset. Furthermore, in Table 1.4 I also provide some statistics by differ-ent quartiles of exposure to the minimum wage, measured by the fractionof workers below the new minimum wage. This Table shows that the firmswho would’ve had to adjust their wages more - in compliance with the pol-icy increase - were less likely to be located in Lima, payed lower wages onaverage, and were smaller and younger. It also shows that highly exposedfirms are more likely to be in the manufacture and services sector.Lastly, Figure A.4 shows the composition of the labor force in Peru in theyears around my analysis. It shows that (i) unemployment was very smalland roughly constant over that time frame; (ii) informal contracts withinformal firms is not that common in this context; and (iii) the informal sectoris very large relative to the formal sector. Additionally, more than half ofthe orange area in this graph corresponds to informal self-employment.1.5 Research Design: FirmsI estimate the employment effects of the minimum wage by comparing theevolution of key outcome variables at firms with many workers affected bythe minimum wage increase to those firms with few affected workers. I followHarasztosi and Lindner (2019) estimate regression models of the followingform:yjt − yj2016yj2016= αt + βtFAj + γtXjt + εjt, (1.5)151.5. Research Design: Firmswhere the left hand side is the percentage change in outcome y relativeto May 2016, the final pre-policy year, and year t. This specification allowstime effects and the impact of firm characteristics, γt, to vary flexibly overtime.I winsorize the percentage changes,yjt−yj2016yj2016, to take values between the1st and 99th percentile in each year. I run the later analysis including versusexcluding firms that shut down in the analysis period, where they experiencea 100 percent decline in their outomes. I measure exposure to the minimumwage, FAj by calculating the fraction of workers for whom the minimumwage increase binds. I predict FAj from the average wage cost and employ-ment counts (observed in PLAME data) to be able to run the analysis usingthe firm-level financial data. I invite interested readers to go to AppendixA.2 for further details of the procedure. This specification above assumesthat the dependent variable is linear in terms of the exposure measure. InFigure A.6 and Figure A.7 I show that the binscatter between such outcomesand the exposure measure are consistent with a linear relationship.I restrict my sample to firms that existed in 2015 and 2016. I estimaterobust standard errors using the logarithm of total employment in 2016 asweights in the regression.12 In my benchmark regression I control for thefollowing variables and their squares: fraction of censored earnings between2015 and 2016; fraction of female workers between 2015 and 2016; and thefraction of university degree holders between 2015 and 2016. In addition, Icontrol for fixed effects for firm age, province, and 1-digit industry.Identification Assumption The regression specification described aboverelies on the assumption that firms with fewer minimum wage workers are avalid estimate of the counterfactual for firms with many affected workers, sothat these firms would follow a parallel trends in the absence of the minimumwage increase. The existence of differential trends before the minimum wagehikes would threaten the credibility of this assumption. Reassuringly, I rejectthe presence of differential trends in the results that will be shown next.12For the EEA data I can also use the logarithm of total revenues in 2015 as weights,and the results remain similar. The reason for using logarithms is that the distributionof employment size and revenues is highly skewed. This can obscure the interpretation ofmy results given that the central limit theorem might not hold in the firm-level weightedregressions(Harasztosi and Lindner, 2019)161.6. Results: Firms1.6 Results: Firms1.6.1 Employment and Wage EffectsEmployment Effects Panel A of Table 1.5 shows the regression resultsfrom Equation (1.5). Columns 1 to 4 show the effects of the minimum wagein the two following years after the minimum wage increase. The pointestimate in column 1 indicates that a year after the minimum wage hikeemployment declines by 9.9 percent (SE 0.7 percent) more at firms where 100percent of the workforce is directly affected by the minimum wage relativeto firms with no exposure. This results include both the extensive margin(firm closure) and intensive margin (layoffs). Column 2 then shows the pointestimates after we control for the set of covariates described in Section 1.5.The point estimate then drops to 7.6 percent (SE 0.8).Columns 3 and 4 repeat the same exercise for the effects two years afterthe minimum wage hike. In particular, Column 4 shows that employment isreduced by approximately 9.5 percent (SE 1.0 percent) lower at firms with100 percent exposure relative to firms with no exposure.Panel (a) of Figure 1.2 shows that there is no evidence of any differentialtrends prior to the minimum wage hike between exposed and non-exposedfirms. In Panel (b) of Figure 1.2 I exclude the extensive margin responses.The disemployment effects are just slightly smaller (5.1 percent decrease, SE0.8 percent) and, again, there is no sign of differential trends among treatedand untreated firms.Wage Effects I now compute the average wage at the firm as the wagecost divided by a statistic of full-time employment at the firm, and use itto construct the dependent variable in Equation (1.5). Therefore, theseregressions unavoidably exclude those firms that closed after the minimumwage increase.Columns 1 and 2 in panel B of Table 1.5 show that the average wage atthe firm increased significantly in the two following years after a minimumwage hike. In particular, column 2 in panel B of Table 1.5 indicates thatthe average wage increases by 11.4 percent (SE 0.4 percent) at firms with100 percent exposure to the minimum wage increase relative to firms withno affected workers.Columns 3 and 4 in panel B of Table 1.5 compute the point estimatestwo years after the minimum wage hike. These columns show that theaverage wage are 11.2 percent (SE 0.4 percent) higher at exposed firmscompared to non-exposed firms. Furthermore, Figure 1.2b shows that there171.6. Results: Firmsis no indication that exposed firms were trending any differently than non-exposed firms prior to the minimum wage increase.In Figure A.8 I compute the implied own-wage elasticity, which is theratio of the employment effects and wage effects, and compare it to recentestimates of this elasticity parameter. To construct standard errors I jointlyestimate both effects and perform a test on the non-linear combination. Theshort-run elasticity is around -0.85 (SE 0.06) one year after the minimumwage increase, and -1.12 (SE 0.08) two years after. The confidence intervalsof these two estimates can reject most of the recent estimates of the own-wage elasticity, which are statistically closer to zero.Heterogeneity and the Informal Sector I then explore heterogeneouseffects across industries, and whether the informal sector has any saliencein explaining this heterogeneity. Figure A.3 plots the point estimates acrossdifferent industries one year after the minimum wage increase. The em-ployment effects are - as expected - more pronounced in the manufacture,commerce , and services sectors. Furthermore, within the construction sec-tor I find employment effects that are statistically indistinguishable fromzero across all years. I also find that largest negative employment effects inthe medium term (3 years after the increase) is in the commerce sector.To test whether the presence of the informal sector plays a role in ex-plaining these differences, I compute a firm-specific measure of exposure tothe informal sector. More precisely, I observe 3-digit occupations in theENAHO dataset, and I compute the share of workers in these occupationsthat work in the informal sector. I classify an occupation as informal if theybelong to the highest quartile of this share 13. Next, I compute the fractionof workers at the firm that belong to an informal occupation, and estimatean interacted version of Equation (1.5):yjt − yj2016yj2016= α0t +α1tFAj + β0t FAj + β1t FIj + βtFAjFIj + γtXjt + ϵjt (1.6)Figure 1.3 plots the employment and wage estimates among firms thatwere highly exposed to the informal sector and those who were not. Inparticular, we can observe that firms that have to compete less for their laborcan increase their wages without having to layoff many workers compared tofirm who have 100 percent of their workforce in those occupations classified13For example, some highly informal occupations are weavers, knitters, and street ven-dors. On the other hand, occupations such as lawyers and doctors are classified as notinformal.181.6. Results: Firmsas informal. In particular, the employment effects are statistically differentbetween firms with high exposure and no exposure to the informal sector(p-value=0.03), whereas the wage effects are not statistically different in theyear after the minimum wage increase (p-value=0.19).1.6.2 Other Firms’ Margins of AdjustmentIn this section, I focus my attention to other firm outcomes that can berelevant for a comprehensive understanding of the effects of minimum wagepolicies on firms.Using the EEA dataset, I leverage the same firm-level approach describedabove. In particular, I estimate Equation (1.5) using the cost of labor,total revenues, profits, and material expenses as dependent variables in theanalysis. For the manufacturing sector, I have information at the productlevel that allows me to compute a price index to use as a dependent variablein these regressions.Effects on Cost of Labor Panel A of Table 1.8 and Table 1.9 showthe main effects on total labor costs. In particular, the Column 1 andColumn 2 in Table 1.9 indicate that surviving firms faced a 12.3% and 21.9%increase in total labor costs during the two years following the minimumwage increase. The last column computes the placebo effect in the yearprior to the increase, which is statistically insignificant. The coefficientsalong with 95% confidence intervals are also shown in Figure 1.9, Panel (a).Effects on Revenue Panel B of Table 1.9 show the coefficients on totalrevenues. Column 1 and Column 2 in Table 1.9 show that firms that didn’tclose after a minimum wage increase had larger revenues in the year afterpolicy enactment (7.4% increase, SE=3.6%). The results of Column 3 showthat there is no differential pre-trends between high- and low-exposed firms.Effects on Material Expenses Panel C of Table 1.8 and Table 1.9 showthat there is no statistically significant evidence that there was a change inmaterial expenses among firms.Effects on Capital Panel D of these tables show some mild evidencethat there could have been some substitution between labor and capital.However, given the statistically significant effects on revenue, wages andemployment, this channel is likely not key to the rest of my analysis.191.7. Research Design: WorkersEffects on Profits I find that profits do not change on average duringthe period of analysis, in response to the minimum wage hike. These resultspotentially mean that firms are not bearing the burden of the increasedlabor costs, but instead passing through the burden onto consumers. Thiswill be further explored in the next section of this paper.Effects on Price Figure 1.10 plots the coefficients of Equation (1.5) us-ing changes of the Laspeyres index as dependent variable. While it seemsto be underpowered, it does suggest that prices increase as a response tothe minimum wage hike. A limitation of this index is that it can only becomputed among products that existed before and after the policy change.Heterogeneity and the Informal Sector Figure 1.11 shows the effectson labor costs and revenue separately for provinces where informality iscommon and provinces where it is not. Panel (a) of Figure 1.11 shows thatlabor costs are statistically closer to zero among firms in highly informallocations. Furthermore, firms in highly informal locations were also lesslikely to increase total revenues. This suggests that the price pass-throughis muted in such cases, consistent with the conceptual framework describedin Section 1.3.1.7 Research Design: WorkersThe effects at the firm-level may not provide a complete picture of theeffects of this policy if workers are able to reallocate across different firms(Dustmann et al., 2022), so that workers can be better off at the expenseof the closure of unproductive firms. In this sense, minimum wage increasescan generate more firm-to-firm transitions in the economy and help workersclimb-up the ladder, thus creating wage spillover effects above the minimumwage (Engbom and Moser, 2022; Berger et al., 2024).In this approach I leverage the fact that I can follow workers over timeand closely follow the empirical strategy described in (Dustmann et al.,2022). Let t∗ = 2016 denote the year of the minimum wage increase. Theidea is to compare outcome changes between two pre-policy periods (t∗ − 2and t∗−1) to outcome changes between the year prior to the minimum wagehike and the year after the increase (t∗ − 1 and t∗) along the distributionof earnings in the baseline period (t∗ − 2 and t∗ − 1, respectively). In thebenchmark specification, I assign workers to 15 small (100 PEN ) earningsbins b (where the first bin b = 1 refers to monthly earnings between 650201.7. Research Design: Workersand 750 PEN, . . . , and the 15th bin to monthly earnings between 2,250 and3,500 PEN ) based on their monthly earnings in (t − 1).14 I then regressindividual outcome changes for both pre-treatment period t = 2016 andpost-treatment t = 2017 using the following specification:yi,t − yi,t−1 =15∑b=1γt,b1{earningsi,t−1 ∈ binb}+ βXi,t−1 + ϵi,t, (1.7)where the parameters γt,b measure the outcome change between t − 1 andt of workers in bin wage b conditional on a vector of individual baselinecharacteristics Xi,t−1. For the post-policy period (2016 vs 2017), the coeffi-cients γt,b capture the effect of the minimum wage across different wage binsb, subject to mean reversion and macroeconomic time effects as potentialconfounders. To eliminate these confounding effects, I subtract the γt,b cor-responding of the pre-policy period (2015 vs 2016). This can be performedby running the following re-parameterized regression:yi,t − yi,t−1 =15∑b=1γ2016,b1{earningsi,t−1 ∈ binb} +15∑b=1δb1{earningsi,t−1 ∈ binb} × POSTt + βXi,t−1 + ϵi,t,(1.8)where δb := γ2017,b − γ2016,b and POSTt indicates the post-treatment yeart = 2017. The coefficients δb trace out the outcome change in the post-policyyears relative to the outcome change in the pre-policy years.Identifying Assumption the parameters δb will capture the causal ef-fect of the minimum wage increase under the assumption that mean rever-sion and macroeconomic time effects are stable over time (i.e. γ2016,b =γ2017,b , b ∈ {1, . . . , 15} in the absence of MW hike). Given that the up-per tail shouldn’t be affected by this policy, I can test whether there wereany macroeconomic time effects in the earnings distribution by investigat-ing whether γ2016,15 = γ2017,15 are close to zero or not. Reassuringly, this isthe case for most of my specifications. If, however, it is the case that theestimated coefficient γ in the last bin is statistically different from zero, I14I pool individuals at the upper tail up to 3,500 PEN to increase precision of myestimates in the control group. The main results are robust to the use of different binwidths.211.8. Results: Workerscan construct generalized difference-in-differences estimates by subtractingγ2017,b=15 from any other bin effect γ2017,b, b ̸= 15. The assumption behindthis approach is that the changes in macroeconomic time effects are constantacross all earnings bins.1.8 Results: Workers1.8.1 Wage, Employment and Reallocation EffectsWage Effects Panel (a) of Figure 1.4 provides evidence that the minimumwage increased earnings of low-wage workers. In particular, this panel plotsone-year earnings growth separately for the years 2015 vs 2016 to 2016vs 2017, as obtained from regressing Equation (1.7) in these two separatesamples. We can observe that low-wage workers typically experience higherwage growth relative to those at the highest part of the earnings distribution.To address the fact that the wage growth at the bottom of the distributionis partly due to macroeconomic time effects and mean reversion, we subtractboth curves. This is systematically performed by regressing Equation (1.8).In panel (b) of Figure 1.4 I plot the point estimates of the differencebetween both curves of panel A. These results show that log earnings in-creased by 0.075 (SE 0.004) for those who earned between 750 and 850 priorto the minimum wage increase. Consistent with the identification strategy,we also observe that the earnings increase at the very top of the distributionis small (-0.01 , SE 0.003). This suggests that macroeconomic conditionswere sufficiently stable during the period 2015-2017. We also observe thatthe minimum wage increase had substantial spillover effects for workers whoearned above the new minimum wage.The estimates are summarized on Panel A of Table 1.6. In particular,Columns 5 and 6 of Table 1.6 compute the difference between the coefficientat different bins and the control group (the highest bin). Reassuringly, thedifference-in-differences estimates remain statistically significant and closerto the baseline results shown in the Figure 1.4.Employment Effects Next, I investigate whether the minimum wageincrease forced workers out of formal employment. To do so, I use theprobability of remaining formally employed as the dependent variable inEquation (1.8). Panel (a) of Figure 1.5 shows that low-wage workers aremuch more likely to exit formal employment compared to those who arehigher in the earnings distribution. Furthermore, the difference betweenthese probabilities between 2015 vs 2016 and 2016 vs 2017 is significantly221.8. Results: Workerssmaller at the top of the earnings distribution. This suggests that, as wewould expect, workers at the top of the earnings distribution have no changein their employment prospects after the minimum wage hike.Panel (c) of Figure 1.5 computes the probability of being employed in trelative to the 2015 vs 2016 period, obtained from Equation (1.8). This fig-ure shows that workers who earned close to the minimum wage experienceda negative impact on their employment prospects. For example, workerswho earned between 750 and 850 were 0.011 pp. (SE 0.005) less likely to beformally employed after the minimum wage increase. Reassuringly, I cannotreject any effect at the top of the earnings distribution, which confirms thatmacroeconomic conditions were stable in this study period.Moreover, Panel B of Table 1.6 show the estimates across different bins.The last three columns compute difference-in-differences estimates by sub-tracting the main effects at different bins to the effect in the control groupReallocation Effects To address whether part of the wage increases cancome from workers climbing up the ladder, I estimate the AKM firm fixedeffects(Abowd et al., 1999) using the time periods prior to the minimumwage increase, and then I compute a change in quality as qk=t−1j(i,t),i − qk=t−1j(i,t−1),i,where qk=t−1j(i,t),i is the time k firm fixed effect of firm j at which worker i isemployed in period t. The advantage of this measure is that it capturescompositional changes rather than mechanical increases in the firm effectsdue to the minimum wage increase. This measure is by construction zeroamong workers who remain at their firm between t− 1 and t, and can onlybe defined among firms that existed in both time periods.Figure 1.6 show that workers who earned slightly above the minimumwage were more likely to transition to higher paying firms and would partlyexplain the presence of spillovers in the wage effects. Additionally, there isno evidence of worker reallocation at the top of the minimum wage, whichreaffirms the assumption of stable macroeconomic effects. On Table 1.7, Ishow the main effects for the bins around the old and new minimum wage.Panel A shows the baseline earnings effects and Panel B shows the coeffi-cients using the change in AKM firm effects, measured prior to the minimumwage hike. In the last two rows of Table 1.7 I compute the percent of theearnings effect due to reallocation. The results suggest that the reallocationchannel was very small in general, which means that most of the earningsgains come mainly from compliance to the new legislation. All in all, theseresults show that in developing economies with low worker productivity wecan expect to see patterns that contrast recent evidence in Dustmann et al.231.8. Results: Workers(2022) for the case of Germany.Heterogeneity and the Informal Sector Figure 1.7 shows the relation-ship between the wage and employment effects and how informal the work-ers’ occupation is. To determine whether an occupation is highly exposed toinformality I compute - using the ENAHO dataset - the share of individualsclassified as informal at each occupation, and then select the highest quartileof this measure. Panel (a) of Figure 1.7 shows that workers at occupationsthat are more common to observe in the informal sector face the largestdisemployment effects, whereas the rest of workers have employment effectsstatistically indistinguishable from zero. On the other hand, Panel (b) ofFigure 1.7 shows that, conditional on remaining employed, workers tend toreceive a wage boost regardless of whether their occupation was classified asexposed to informality or not.1.8.2 Do workers exit the formal sector?The evidence gathered up to this point suggests that workers exit formalemployment after a minimum wage increase, especially at occupations com-monly observed in the informal sector. A possible explanation of theseresults could be that firms switch workers under a formal contract to be-ing hired informally (i.e. informal employment within the formal sector).In this sub-section I show evidence that the size of the formal sector is in-deed smaller after the minimum wage increase, suggesting that workers arereallocating towards the informal sector.First, in Figure 1.8 I check using employment counts whether the numberof workers with a formal contract decreases among individuals who earnedclose to the new minimum wage. Panel (a) of Figure 1.8 shows the em-ployment counts two years after the minimum wage hike (2018) and thebaseline year (2016). Next, Panel (b) of Figure 1.8 computes the differencein each bin, as well as the cumulative difference. This Figure shows thatthere is a missing mass of workers below the new minimum wage that isnot compensated by the new mass above the minimum wage. This indicatesthat workers are indeed exiting formal employment. A potential explana-tion to this is that workers remain at the same firm but now hired withouta formal contract. I then turn to the ENAHO dataset to test this whetheremployment in the formal sector decreases.I follow the approach described in Giupponi et al. (2024) that combinesregional variation with a frequency distribution approach. The idea is toselect high-wage provinces and use them as a control group, and then com-241.9. Conclusionspare workers with the same skill level - defined as wages net of time andplace effects - across high- and low-wage provinces. In other words, thismethodology is based on the fact that workers with the same skill would’veearned the same had they lived in the same location. I invite interestedreaders to check any details of the implementation of this methodology inSection III.B in Giupponi et al. (2024).Figure 1.12 shows the results of applying this methodology on the en-tire formal sector observed in the ENAHO dataset. In particular, Panel (a)shows the distribution of province fixed effects that comes from a Mincer-style regression of wages against individual characteristics, and fixed effectsof location and year. The individual characteristics include a quadratic poly-nomial of age, a female indicator, educational level, industry sector, and oc-cupation. Panel (b) of Figure 1.12 shows the employment percent changesat 100 PEN bins relative to the new minimum wage. The change in employ-ment below the new minimum wage (∆b) is equal to -3.10% (SE=0.53%),and the cumulative employment percentage change up to 11 bins relativeto the new minimum wage is also negative and larger. This indicates thatthere was a statistically significant employment decrease in the formal sectoramong low-wage workers.1.9 ConclusionsIn this chapter, I presented novel empirical evidence of the role of informalself-employment in explaining the heterogeneous effects of the minimumwage in countries with a large informal sector.I find that, consistent with the theoretical model of discrete choice involv-ing formal versus informal sector choice and oligopsonistic firm competition,the presence of informal self-employment can shape the effects of the min-imum wage. After the minimum wage increase, I find that (i) firms whowere more exposed to presence of the informal sector - either in terms ofoccupational composition or geographic location - exhibit larger disemploy-ment effects, and (ii) workers were more likely to exit formal employmentaltogether.My findings also indicate that - on average - surviving firms didn’t expe-rience profit losses, as their revenues increased along with total labor costs.Additionally, I find that low-wage workers that remained in the formal sectorexperienced significant gains in terms of earnings up to the 80th percentile ofthe earnings distribution. These results motivate an analysis of the incidenceof the minimum wage that will be explored in the upcoming chapter.251.9. ConclusionsThe next chapter is devoted to perform a redistribution analysis of thispolicy through the study of changes in purchasing power among differentindividuals and households.261.10. Figures1.10 FiguresFigure 1.1: Minimum Wage to Median EarningsNote: The blue line shows the evolution of the ratio of the minimum wage to median earningsfrom 2015 to 2019. The dashed lines show, as a reference, the ratio of minimum wage to medianearnings in Brasil and the US in 2012. Source: PLAME.271.10. FiguresFigure 1.2: Employment and Wage Effects: Firm Approach(a) Employment Effects(b) Wage EffectsNote: This figure shows the estimates from Equation (1.5) using firm employment and averagewage as the outcome variables, respectively. Panel (a) plots the estimates on total employment,and Panel (b) plots the estimates on the average wage at the firm, defined as the total wage costdivided by the FTE statistic. Regressions are weighted by the logarithm of total employmentin 2016, and robust standard errors are computed. Point estimates and their 95% confidenceintervals are shown in this figure. The vertical red line corresponds to the baseline year, prior tothe minimum wage hike. Source: PLAME.281.10. FiguresFigure 1.3: Employment and Wage Effects, by Informality(a) Employment Effects(b) Wage EffectsNote: This figure shows the estimates from Equation (1.6) using firm employment and averagewage as the outcome variables, respectively. The darker line represents the estimates amongfirms who were highly exposed (in terms of occupational composition) to the informal sector,whereas the lighter line represents the effect among firms with low exposure. Regressions areweighted by the logarithm of total employment in 2016, and robust standard errors are computed.Point estimates and their 95% confidence intervals are shown in this figure. The vertical red linecorresponds to the baseline year, prior to the minimum wage hike. Furthermore, the p-value ofthe difference between the dark and light lines are shown. Source: PLAME.291.10. FiguresFigure 1.4: Earnings Effects: Worker ApproachNote: This figure plots the wage effects under the worker approach described in Section 1.7. Thedependent variable is the logarithm of monthly earnings. The upper panel plots the coefficientsfrom Equation (1.7) as of February 2016 (pre-policy period) and February 2017 (post-policy pe-riod). The lower panel shows the difference between the blue and red line in the upper panel,which correspond to the coefficients of Equation (1.8). Point estimates and their 95% confidenceintervals are shown in this figure. Standard errors are clustered at the province level. Source:PLAME.301.10. FiguresFigure 1.5: Employment Effects: Worker ApproachNote: This figure plots the employment effects under the worker approach described in Section1.7. The dependent variable is an indicator equal to 1 if the worker remains employed. Theupper panel plots the coefficients from Equation (1.7) as of February 2016 (pre-policy period) andFebruary 2017 (post-policy period). The lower panel shows the difference between the blue andred line in the upper panel, which correspond to the coefficients of Equation (1.8). Point estimatesand their 95% confidence intervals are shown in this figure. Standard errors are clustered at theprovince level. Source: PLAME.311.10. FiguresFigure 1.6: Reallocation Effects: Worker ApproachNote: This figure plots the reallocation effects using the worker approach described in Section1.7. The dependent variable is the change in firm AKM fixed effects, which is measured priorto the minimum wage increase. The upper panel plots the coefficients from Equation (1.7) asof February 2016 (pre-policy period) and February 2017 (post-policy period). The lower panelshows the difference between the blue and red line in the upper panel, which correspond to thecoefficients of Equation (1.8). Point estimates and their 95% confidence intervals are shown inthis figure. Standard errors are clustered at the province level. Source: PLAME.321.10. FiguresFigure 1.7: Employment and Wage Effects, by Informality(a) Employment(b) EarningsNote: This figure plots the employment and wage effects using the worker approach described inSection 1.7, respectively. The upper panel plots the coefficients from Equation (1.8) for workersin occupations that are highly versus lowly exposed to informality separately. The dependentvariable used is an indicator that equals 1 when the worker remains employed. The lower paneldoes the same exercise with the logarithm of earnings as the dependent variable. An occupationis defined as highly exposed to informality if it belongs to the top quartile of share of informalemployment for that occupation (3-digits). Point estimates and their 95% confidence intervals areshown in this figure. Standard errors are clustered at the province level. Source: PLAME.331.10. FiguresFigure 1.8: Frequency of earnings distribution in 2016 and 2018(a) Employment counts(b) Cumulative differenceNote: This figure shows the employment counts along the distribution of log real earnings (ad-justed by cpi) in the upper panel. The lower panel then shows the difference between the countsin 2016 and 2018 . Negative values on the blue bars indicate that there is a missing mass ofworkers at those bins. The red line in the lower panel indicates the cumulative difference. Source:PLAME.341.10. FiguresFigure 1.9: Firms’ Margins of Adjustment(a) Labor Costs(b) Total Revenue(c) Material ExpensesNote: This figure plots the effects of the minimum wage on some firm-level measures based onEquation (1.5). These regressions are run on the set of firms that existed throughout the entireperiod of analysis (i.e., it excludes closures). Panel (a) shows the effects on total labor costs;Panel (b) shows the effects on total revenue; and Panel (c) shows its effects on material expenses.Regressions are weighted by the logarithm of total employment in 2016, and robust standard errorsare computed. Point estimates and their 95% confidence intervals are shown. Source: EEA.351.10. FiguresFigure 1.10: Effect on Laspeyres Price Index in ManufactureNote: This figure shows the coefficients of Equation (1.5) where the dependent variable is aLaspeyres price index constructed for the manufacture sector. These regressions, by construction,can only be computed on the set of products that exist within any two consecutive years. Re-gressions are restricted to firms that existed throughout the entire period of analysis, weightedby the logarithm of total employment in 2016, and robust standard errors are computed. Pointestimates and 95% confidence intervals are shown. Source: EEA.361.10. FiguresFigure 1.11: Effects on Labor Cost and Revenue, by Informality(a) Labor Cost(b) Total RevenueNote: This figure plots the effects of the minimum wage on total labor costs and total revenue,separately for firms located in provinces with high exposure to informality versus provinces withlow exposure. A highly exposed province is defined as a province in the highest quartile of theshare of informal employment across provinces. Regressions are weighted by the logarithm of totalemployment in 2016, and robust standard errors are computed. Point estimates and their 95%confidence intervals are shown. Source: EEA.371.10. FiguresFigure 1.12: Aggregate Employment Change in the Formal Sector(a) Distribution of province-level premia(b) Employment changes in the formal sectorNote: This figure shows the approach described in Giupponi et al. (2024) to estimate employ-ment changes along the wage distribution. The upper panel shows the distribution of province-levelpremia, measured by province fixed effects. High paying provinces are thus selected as a controlgroup. The lower panel shows the employment changes in the earnings bins relative to the mini-mum wage. The 0 in the X-axis indicate the earnings bin corresponding to those earning the oldminimum wage. Source: ENAHO.381.11. Tables1.11 TablesTable 1.1: Who are the minimum wage workers?Earnings bin in 2016 (PEN)[650, 850] (850, 1250] (1250, 3250]Lima 0.14 0.18 0.18Female 0.42 0.38 0.27By educationShare low skilled 0.05 0.05 0.03Share medium skilled 0.48 0.50 0.44Share high skilled 0.47 0.45 0.53By ageShare less than 24 0.09 0.08 0.02Share 24-44 0.63 0.72 0.76Share 45-65 0.27 0.19 0.22By contractPart-time (PT) 0.02 0.01 0.01Permanent (P-FT) 0.36 0.22 0.30Temporary (T-FT) 0.62 0.77 0.69By firm size1-4 employees 0.06 0.03 0.035-19 employees 0.11 0.09 0.0920-49 employees 0.06 0.06 0.0650+ employees 0.77 0.82 0.82By industry structureAgriculture; mining 0.09 0.06 0.06Manufacturing; electricity; waste management 0.11 0.13 0.14Construction; wholesale and retail 0.16 0.15 0.17Transportation; accommodation; food services 0.09 0.09 0.09Information; finance; real estate 0.04 0.11 0.11Professional, administrative, support services 0.12 0.10 0.10Public administration; education; human health 0.03 0.04 0.05Arts, entertainment; other services 0.36 0.31 0.28Number of observations 324,134 488,347 834,485Note: This table shows the characteristics of individuals who were below the new minimum wagein 2016 (i.e. minimum wage workers), and workers higher up in the earnings distribution. Source:PLAME.391.11. TablesTable 1.2: Firm characteristics in PLAME datasetAll Manufacturing Commerce Services(1) (2) (3) (4)Avg Wage (PEN) Mean 1,632.22 1,540.34 1,726.81 1,155.93Number of Workers Mean 115.29 135.10 89.81 59.98Firm Age Mean 13.22 15.91 12.44 10.15Female Share Mean 0.33 0.29 0.36 0.39Avg Age (Workers) Mean 39.59 39.62 38.80 37.28Fraction Affected Mean 0.32 0.34 0.31 0.49p5 0 0 0 0p25 0.04 0.05 0.02 0.04p50 0.17 0.19 0.16 0.5p75 0.60 0.67 0.6 0.8p95 1 1 1 1Observations 23,477 4,520 8,007 2,260Note: This table shows some characteristics across different firms in the PLAME dataset. Thefirst row compares the average wage at the firm, which is defined as the wage bill divided byemployment (FTE). The second row computes the number of workers at the firm. The thirdand fourth row compare the firm age (in years) and the share of female employment at the firm,respectively. The fifth row compares the average age of the firm workers. Finally, the sixth rowand onwards compute the fraction affected, defined as the share of employment who earned belowthe new minimum wage of 850 PEN. Source: PLAME.401.11. TablesTable 1.3: Firm characteristics in EEA datasetFirm Characteristics 2015 2016 2017Mean St.d. Mean St.d. Mean St.d.Firm Age 18.79 14.61 17.69 14.72 19.00 14.84Number of Establishments 5.06 19.93 4.44 13.00 4.96 15.56Number of Executives 4.53 16.99 4.17 10.17 4.68 11.41Number of Perm Employees 93.75 424.22 97.48 386.31 121.19 534.00Number of Non-Rem Emp 1.25 8.82 1.32 9.61 1.84 20.79Number of Third-Party Emp 4.06 42.38 4.43 73.19 2.94 32.91Wage Bill (106 PEN) 10.10 20.81 9.53 20.43 10.90 20.82Value Added (106 PEN) 22.80 86.70 22.00 83.20 24.00 102.20Net Revenue (106 PEN) 5.06 19.93 4.44 13.00 4.96 15.56Number of Observations 4,901 5,319 4,781Note: This table shows the mean and standard deviation of some firm characteristics in theEEA dataset from year 2015 to 2017. Source: EEA.411.11. TablesTable 1.4: Who are the exposed firms?Quartiles of FAj in 2016Quartile 1 Quartile 2 Quartile 3 Quartile 4Lima 0.47 0.41 0.38 0.39Avg Wage (PEN) 2480.81 1693.09 1170.64 888.55Number of Workers 129.60 157.78 127.36 16.20Firm Age 16.03 14.44 11.53 9.39Manufacture 0.18 0.21 0.19 0.22Commerce 0.35 0.31 0.30 0.31Services 0.04 0.06 0.12 0.15Observations 5,875 5,908 5,849 5,845Note: This table shows some firm characteristics in the EEA dataset across differentquartiles of the fraction affected variable in 2016. The second column shows the averagecharacteristics of firms who were less exposed to a minimum wage increased, and thelast column shows the characteristics of firms who were the most exposed. Source:EEA.421.11. TablesTable 1.5: Employment and Wage EffectsMain Main Placebochanges between changes between changes between2016 and 2017 2016 and 2018 2016 and 2015(1) (2) (3) (4) (5) (6)Panel A. Change in firm-level employment (including closures)Fraction affected -0.099 -0.076 -0.138 -0.095 0.002 0.013(0.007) (0.008) (0.007) (0.010) (0.005) (0.006)Constant -0.030 -0.045 0.002(0.003) (0.005) (0.002)Observations 23,488 21,088 23,488 21,088 23,477 21,081Employment elasticity with -0.75 -0.58 -1.06 -0.73respect to MW (directly affected) (0.05) (0.06) (0.07) (0.08)Panel B. Change in firm-level employment (excluding closures)Fraction affected -0.057 -0.051 -0.088 -0.072 0.002 0.013(0.006) (0.008) (0.008) (0.009) (0.005) (0.006)Constant 0.008 0.024 0.002(0.006) (0.004) (0.002)Observations 22,073 19,902 21,208 19,173 23,477 21,081Employment elasticity with -0.44 -0.39 -0.68 -0.55respect to MW (directly affected) (0.04) (0.05) (0.06) (0.07)Panel C. Change in firm-level average wageFraction affected 0.116 0.114 0.113 0.112 0.031 0.031(0.003) (0.004) (0.003) (0.004) (0.001) (0.001)Constant 0.041 0.075 -0.024(0.001) (0.002) (0.001)Observations 22,073 19,902 21,208 19,173 23,477 21,081Employment elasticity with -0.85 -0.59 -1.22 -0.74respect to wage (0.06) (0.08) (0.09) (0.11)Controls No Yes No Yes No YesNote: This table shows the main effects on wages and employment at the firm-level. Panel A and B computethe employment effect including and excluding the extensive margin, respectively. Panel C computes the effecton the firm-level average wage, defined as the wage bill divided by FTE employment. The own-wage employmentelasticity is then computed by dividing the results from Panel A and panel C. The set of controls include somecharacteristics measures in 2016: the fraction of censored workers’ earnings, the fraction of females, fraction ofcollege graduates, and fixed effects of firm age, province, and industry sector. Regressions are weighted by thelogarithm of employment in 2016. Source: PLAME.431.11.TablesTable 1.6: Effects of the Minimum Wage on Wages and Employment: Individual ApproachChanges relative to 2015 versus 2016 Difference-in-differences(1) (2) (3) (4) (5) (6) (7)Bin at t− 1 [650, 750) [750, 850) [850, 950) [2250, 3500] (1) - (4) (2) - (4) (3) - (4)Panel A. Earnings2016 versus 2017 0.0955 0.0751 0.0441 -0.0104 0.1060 0.0855 0.0546(0.0050) (0.0042) (0.0038) (0.0029) (0.0038) (0.0040) (0.0042)Baseline change (2015 versus 2016) 0.0383 0.0410 0.0383 -0.1253(0.0212) (0.0224) (0.0232) (0.0252)Panel B. Employment (1 if remain employed)2016 versus 2017 -0.0127 -0.0115 -0.0131 -0.0000 -0.1268 -0.0114 -0.0131(0.0023) (0.0049) (0.0071) (0.0054) (0.0050) (0.0040) (0.0055)Baseline change (2015 versus 2016) 0.4196 0.4318 0.4378 0.5374(0.0150) (0.0145) (0.01523) (0.01638)Note: This table shows robustness checks. The dependent variable in Panel A is an indicator of whether the head of household’s spouse isworking (employee or self-employed); in Panel B is an indicator of whether the head of household’s spouse is working (employee only); PanelC is an indicator of whether head of household’s spouse is an unpaid family worker; Panel D is an indicator of whether all teens living in thehousehold go to school; and Panel E uses an indicator of whether all teens go to school while they don’t work at all. All regression controlsfor age, education, gender, and district fixed effects in the baseline period t − 1. Robust standard errors are reported in parentheses. Source:PLAME.441.11. TablesTable 1.7: Reallocation Effects of the Minimum Wage: Individual ApproachMain effects (2016 vs 2017)Earnings Bin at t− 1 [650, 750) [750, 850) [850, 950)(1) (2) (3)Panel A. EarningsEstimated effect 0.1060 0.0855 0.0546(0.0038) (0.0040) (0.0042)Panel B. Firm’s AKM effectsEstimated effect -0.0037 -0.0008 0.00167(0.0011) (0.0019) (0.0017)Earnings growth due to reallocation (% of total effect)Calculated using all switchers 0.00 0.00 3.05Calculated using upward switchers 2.15 4.70 5.21Note: This table shows the reallocation effects of the minimum wage using the individual-level design.Panel A computes the main treatment effects on earnings. Panel B compute the effect on firm premia,measured by the AKM firm fixed effects. By dividing the effect on firm premia with respect to the effecton earnings I can compute the earnings growth due to reallocation. Source: PLAME.451.11. TablesTable 1.8: Firms’ Margins of Adjustment (incl. closures)Main Main Placebochanges between changes between changes between2015 and 2016 2015 and 2017 2015 and 2014(1) (2) (3)Panel A. Change in total labor costFraction affected -0.495 0.185 0.031(0.069) (0.075) (0.022)Panel B. Change in revenueFraction affected -0.504 0.055 0.011(0.064) (0.066) (0.027)Panel C. Change in materialsFraction affected -0.446 -0.152 -0.147(0.140) (0.152) (0.151)Panel D. Change in capitalFraction affected 0.135 0.165 0.006(0.068) (0.087) (0.055)Panel E. Change in profits (relative to revenue in 2015)Fraction affected -0.043 -0.008 -0.008(0.003) (0.010) (0.008)Observations 4,343 4,343 4,343Controls Yes Yes YesNote: This table shows the effects on labor costs, revenue, materials, capital and profits. This tablerestricts the sample to exclude firms that closed after the minimum wage increase. The first two columnsoutline the effects one and two years after the policy, respectively. In Column 3 I compute a placeboeffect between 2015 and 2014. The set of controls include quadratic polynomials of the following variables:average profitability between 2014 and 2015, average wage between 2014 and 2015, average depreciationrate between 2014 and 2015, and average labor share of income. It also includes fixed effects of sector,region, firm age, and type of firm. Regressions are weighted by the logarithm of employment in 2015.Source: EEA.461.11. TablesTable 1.9: Firms’ Margins of Adjustment (excl. closures)Main Main Placebochanges between changes between changes between2015 and 2016 2015 and 2017 2015 and 2014(1) (2) (3)Panel A. Change in total labor costFraction affected 0.123 0.219 0.031(0.039) (0.051) (0.022)Observations 3,440 3,185 4,343Panel B. Change in revenueFraction affected 0.074 0.051 0.011(0.036) (0.046) (0.027)Observations 3,440 3,185 4,343Panel C. Change in materialsFraction affected 0.079 -0.140 -0.147(0.152) (0.176) (0.151)Observations 1,995 1,796 2,428Panel D. Change in capitalFraction affected 0.135 0.165 0.006(0.068) (0.087) (0.055)Observations 3,440 3,185 4,343Panel E. Change in profits (relative to revenue in 2015)Fraction affected -0.002 -0.006 -0.008(0.010) (0.012) (0.008)Observations 3,440 3,185 4,343Controls Yes Yes YesNote: This table shows the effects on labor costs, revenue, materials, capital and profits. This tablerestricts the sample to exclude firms that closed after the minimum wage increase. The first two columnsoutline the effects one and two years after the policy, respectively. In Column 3 I compute a placeboeffect between 2015 and 2014. The set of controls include quadratic polynomials of the following variables:average profitability between 2014 and 2015, average wage between 2014 and 2015, average depreciationrate between 2014 and 2015, and average labor share of income. It also includes fixed effects of sector,region, firm age, and type of firm. Regressions are weighted by the logarithm of employment in 2015.Source: EEA.471.11. TablesTable 1.10: Effect on Firm-Level Price Index in the Manufacturing SectorExists betweenAll firms 2015 and 2018(1) (2) (3) (4)Panel A. Change between 2015 and 2016 (short-term)Fraction affected -0.006 0.086 0.040 0.038(0.133) (0.204) (0.091) (0.174)Constant 1.108 0.796 1.088 1.153Observations 531 530 368 368Panel B. Change between 2015 and 2017 (medium-term)Fraction affected 0.010 0.155 0.057 0.384(0.114) (0.175) (0.134) (0.192)Constant 2.089 2.322 2.082 2.515Observations 325 325 275 275Controls No Yes No YesNote: This table shows the price effect in the manufacturing sector. The first two rowscompute the estimates using all firms and the last two rows restrict the analysis to firmsthat existed throughout 2015 to 2018. The set of controls include quadratic polynomials ofthe following variables: average profitability between 2014 and 2015, average wage between2014 and 2015, average depreciation rate between 2014 and 2015, and average labor share ofincome. It also includes fixed effects of sector, region, firm age, and type of firm. Regressionsare weighted by the logarithm of employment in 2015. Source: EEA.48Chapter 2A redistribution analysis ofthe 2016 Peruvian minimumwage increase2.1 IntroductionThis chapter is devoted to studying whether the consequences of minimumwage hikes benefits or harms workers and households. By using some of theempirical results in the previous chapter, I compute changes in purchasingpower (income changes versus expense changes) to perform a redistributionanalysis of the minimum wage increase in Peru that took place in 2016.First, I study the incidence of the minimum wage on consumers andfirms. I find that, among firms that do not close after the minimum wagehike, the burden of the increased labor costs lies almost completely on theconsumers. Furthermore, I find that rich households are the ones that con-sume a higher share of goods produced using minimum wage workers 15.By combining these results with detailed consumption information in theENAHO dataset, I compute the expected price and expense increase in anindividual- and household-level analysis.Next, by using the reduced form estimates from the worker-level ap-proach, I find that the increase in expected income - that is, accounting forthe fact that workers could exit formal employment after a minimum wagehike and receive income from this outside option - is positive for low-wageworkers. In addition, workers in the informal sector do not seem to perceiveany income increases due to this policy.Among formal workers below the 80th percentile of the wage distribu-tion, the positive income effect dominates the increase in expected expensesfrom the price pass-through of the minimum wage. Workers beyond thispercentile perceive no boost in earnings and, additionally, bear the burden15This is a qualitatively different result to what you find in developed countries. See,e.g., Harasztosi and Lindner (2019) for the case of Hungary.492.2. Dataof paying for most of the pass-through to prices. Finally, I show that lowerincome households do not perceive any income increases due to their em-ployment composition. These households are mostly composed by informalworkers and thus there is no redistribution from rich to poor households.This highlights that the presence of the informal sector poses difficulties ofminimum wages to generate redistribution, in addition to limited efficiencygains explored in Chapter 1.These results contribute to existing literature on the idea that inflationdoes vary across the income distribution (Jaravel, 2021). Furthermore, thischapter proposes a simple method to approximate income-specific changesin purchasing power that do not rely on potentially strong assumptions onhousehold preferences (Atkin et al., 2024).2.2 DataThe data used in this chapter follow closely the one described in Section 1.4.In this chapter, I augment data used for an individual and household levelanalysis, which is described next.2.2.1 Individual and Household Panel DataThe ENAHO dataset has a rotating panel that follows households over time.In particular, I use the panel sample of households followed between 2014and 2016, consisting of approximately 3,000 households and their householdmembers. Therefore, I also observe a set of individuals followed throughoutthat same period of time.Among these individuals and households, ENAHO provides detailed ex-penses by all household members on different goods and services, and followsthe path of individuals who were informal before the minimum wage hike.Both of these features will be key for the upcoming analysis. Furthermore,within the food consumption category, ENAHO provides information onprices and quantities, which allows me to compute a Laspeyres price index.For the household level analysis I exclude households that report havingzero work-related income between 2014 and 2015, and those households withtop and bottom 1% changes in terms of total income and savings growthbetween 2014 and 2015. My final sample includes 2,215 households followedfrom 2014 to 2016, which represent approximately 9,000 individuals on av-erage per year.502.3. Research Design: Informal Workers2.3 Research Design: Informal WorkersI follow a similar strategy than the one described in Section 1.7. In partic-ular, using the subset of informal workers, I assign each individual to eightincome bins b based on their monthly income in (t − 1). The bins havetwice the width as in Section 1.7, and are given by [450, 650), [650, 850),[850, 1050), · · · , [1650, 1850), and beyond 1850 PEN. I assign wider bins rel-ative to Section 1.7 due to a small size of observations within some bins inthe ENAHO dataset when using smaller bins. While the results are quali-tatively the same, the coefficients are much noisier.Next, I estimate the following regression:yi,t − yi,t−1 =8∑b=1γ2016,b1{incomei,t−1 ∈ binb} +8∑b=1θb1{incomei,t−1 ∈ binb} × POSTt + βXi,t−1 + ϵi,t, (2.1)where θb := γ2017,b − γ2016,b and POSTt indicates the post-treatment yeart = 2017. In other words, the coefficients θb trace out the outcome change inthe post-policy years relative to the outcome change in the pre-policy years.Identifying Assumption the parameters θb will capture the causal ef-fect of the minimum wage increase under the assumption that mean rever-sion and macroeconomic time effects are stable over time (i.e. γ2016,b =γ2017,b , b ∈ {1, . . . , 8} in the absence of MW hike). Given that informalworkers at the upper tail shouldn’t be affected by this policy, I can testwhether there were any macroeconomic time effects in the earnings distri-bution by investigating whether γ2016,8 = γ2017,8 are close to zero or not.2.4 Research Design: HouseholdsI compute the fraction of workers below the new minimum wage in PLAMEat the regional level (more precisely, at the departamento geographical level)16.I then estimate regressions of the form:yht − yj2015yj2015= αt + βtFADepartment(h) + γtXht + εht, (2.2)16These delimitations are effectively state-level. Peru has 25 of these geographical units.512.5. The Incidence of Minimum Wage in Peruwhere the left hand side is the percentage change in outcome y relative toyear 2015, the final pre-policy year, and year t. This specification allowstime effects and the impact of household characteristics, γt, to vary flexiblyover time.I winsorize the percentage changes, yht−yh2015yh2015 , to take values betweenthe 1st and 99th percentile in each year. I restrict my sample to house-holds followed from 2014 to 2016. I estimate robust standard errors usingthe logarithm of total number of household members in 2015 as weightsin the regression. In my benchmark regression I control for the followingvariables and their squares: fraction of female household members between2014 and 2015, fraction of university degree holders between 2014 and 2015.In addition, I control for fixed effects for macro-region (northern coast, cen-tral coast, southern coast, northern highlands, central highlands, southernhighlands, rainforest area, and metropolitan Lima).Identification Assumption The regression specification described aboverelies on the assumption that households in regions with low exposure to theminimum wage are a valid counterfactual for households in highly exposedregions, so that these households would follow a parallel trends in the absenceof the minimum wage increase. The existence of differential trends beforethe minimum wage hikes would threaten the credibility of this assumption.Reassuringly, I reject the presence of differential trends in the upcomingresults.2.5 The Incidence of Minimum Wage in PeruIn this section I first explore which households are the ones that consumemore of the goods produced by minimum wage workers. These results set thestage to then decompose the incidence of the minimum wage on consumersand firm owners to estimate how much expenses change across differentworkers and households after the minimum wage hike.2.5.1 Who pays for the minimum wage?I follow Macurdy (20115) and Harasztosi and Lindner (2019) and computehow much each industry s is exposed to the MW. By using input-outputtables for Peru, where B(i, j) denote share of commodity j produced byindustry s, and U(i, j) denote share of commodity j used by industry s, I522.5. The Incidence of Minimum Wage in Perucomputees = (I−BU)−1BwagebillMWswagebills× 23, (2.3)where the term 2/3 corresponds to the participation of labor in production.Next, using consumption information in the ENAHO dataset, I matchevery product to a particular industry that produces it 17. I then computethe following measure:Sh. cons produced by MW workers =∑sshare of expenses in s× esFigure 2.1 shows a binscatter between this measure and household in-come distribution. A somewhat unexpected result is that richer householdstend to have a higher share of consumption produced by minimum wageworkers compared to poor households. This result implies that if consumersbear the burden of the increases in labor costs, it will most likely rest onthe shoulders of richer individuals.2.5.2 Decomposing the change in labor costThe results in this section are built on the following identity:Profit ≡ Revenue−Material− LaborCost−Depr−MiscItems,where Depr represents depreciation costs, and MiscItems represents othermiscellaneous expenses. The identity implies the following expression:∆LaborCostRevenue2016=∆RevenueRevenue2016− ∆MaterialRevenue2016− ∆MiscItemsRevenue2016︸ ︷︷ ︸Consumers Pay(2.4)− ∆DeprRevenue2016− ∆ProfitRevenue2016︸ ︷︷ ︸Firm Owners Pay.I estimate Equation (1.5) on each term of the previous expression todecompose the incidence of the minimum wage. Table 2.1 shows the esti-mates of this approach. Column 2 of this Table shows that, one year postpolicy, the burden lies almost entirely on consumers (98.35% of labor costsare financed by consumers). Two years after the minimum wage hike thispercentage falls to 61%, approximately.17For example, if a household reports to have bought a TV, I assign that expense tomanufacturing sector.532.6. The Gains of the Minimum Wage in Peru2.5.3 Estimating the Change in ExpensesThe approach I will follow to estimate how much expenses change for a givenindividual or household (∆E) is to compute the following product∆E =Pass-through of Labor Cost onto Price× Change in Labor Cost× Total Expense in MW-produced Goods.The product of the first two terms in right hand side of the previousexpression represent the change in prices due to the minimum wage. Mybenchmark estimate of the Pass-through of Labor Cost onto Price is 0.98,as it was obtained in the previous subsection. Furthermore, the results inChapter 1 provide an estimate of the Change in Labor Cost given by 0.123.These imply a baseline estimate of the price effect of 0.98× 0.123 = 0.12.In addition, I use the effects on the Laspeyres price index for the man-ufacturing sector computed in Section 1.6.2 of Chapter 1 as an alternativemeasure of the price effect. This measure is rather noisy, and will use it asa referential upper bound. Lastly, I construct a food price Laspeyres indexusing consumption information in the panel version of ENAHO from 2014to 2016. I use this index as a dependent variable in Equation (2.2). Theresults are shown in Figure B.1. This graph indicates that one year after theminimum wage was raised, food consumption prices rose by roughly 6.5%.This increase is larger than the annual inflation of that year of roughly 3.3%.2.6 The Gains of the Minimum Wage in PeruTo estimate the expected change in income (∆I) for different individuals andhouseholds I need to account for the employment effects that, as discussedin Chapter 1, reallocate workers towards the informal sector.My benchmark approach is to analyze these changes within the differentbins defined in Section 1.7. In particular, for a formal worker at bin b andearning wb on average, I compute:∆Ibwb:=∆wbwb︸ ︷︷ ︸Gain from MW if stay+(winfb − wb)wb∆Pb(leave)︸ ︷︷ ︸Earn at informal sector if leave, (2.5)where ∆wbwb and ∆Pb(leave) are proxied by the estimates obtained in Figures1.4 and 1.5, respectively. Furthermore, winfb is a measure of how much a542.7. A Redistribution Analysis of the Minimum Wage in Peruparticular worker would earn if they transition from the formal to the in-formal sector. In the panel version of ENAHO, I can compute on averagehow much a worker at a particular bin b loses if they transition towards theinformal sector.I jointly estimate the elements of Equation (2.5) and bootstrap standarderrors. The resulting expected gains at each bin are shown in Figure 2.2. Inparticular, I find that low-wage formal employees have gains well beyond theold and new minimum wage despite having a greater risk of being displacedout of the formal sector. Part of the reason for this result is that these typeof workers tend to earn similar amounts (despite being slightly smaller) towhat they previously earned in the formal sector.An alternative approach to compute the expected gains is described inAppendix B.2, although the results are qualitatively similar.2.7 A Redistribution Analysis of the MinimumWage in PeruMy estimates provide sufficient information to perform a redistribution anal-ysis of the minimum wage under the following assumptions:1. No substitution away from goods produced by minimum wage workersas a response to the minimum wage hike.2. There is no income effect in the informal sector.Reassuringly, Figure 2.3 shows the estimates from Equation (2.2) ondifferent consumption categories. The dependent variables in these regres-sions are the changes in total household income and consumption categoriesrelative to the total household income in 2015. These results suggest nosubstitution across consumption categories as a response to the minimumwage increase.Additionally, Figure 2.5 shows the estimates from the design described in2.3. I show that there is no evidence that the income of incumbent informalworkers increased after the minimum wage hike. These previous resultsfurther support the approach following next.2.7.1 Redistribution: Individual AnalysisUsing the ENAHO, I obtained at the household level the share of consump-tion in goods produced by minimum wage workers. Next, I compute the552.7. A Redistribution Analysis of the Minimum Wage in Peruaverage per-capita expenses, and the average share of consumption in goodsproduced by minimum wage workers at each wage bin b.Table 2.2 shows the implied changes in income and expenses for workersat different earnings bins. I find that low-wage workers are better off afterthe minimum wage increase. The opposite is true for high-wage individu-als, as they bear the burden of the increased labor costs of firms and theydon’t perceive any spillover gain in terms of income. For example, workerswho earned between the old and new minimum wage were expected to gain65 PEN on average, and increase their expenses by 11.8 PEN. Everythingconsidered, these results suggest that low-wage formal workers are the clearwinners from this policy change.2.7.2 Redistribution: Household AnalysisTo perform a similar exercise on households, I compute the average gainof minimum wage workers as the average gain in the first 5 earnings binsestimated in Figure 2.2:∆IMWIMW≡ 155∑b=1∆Ib/wb.Next, I multiply the gain defined above by the average household in-come at decile d, denoted as I lMW,d, to obtain the expected income change.The expected changes in expenses are computed in a similar fashion to theprevious section, where I now compute the average share of consumption ofgoods produced by minimum wage workers at each household decile. I thenmultiply that number by the household-level total expenditures to obtainthe (average) total expenditures in production by minimum wage workers.Table 2.3 shows that poor households have very few members engaged informal employment and are thus not perceiving income gains. In particular,households at the first two deciles in the household income distribution haveless than 3% of their workers under formal employment. As a matter of fact,minimum wage workers are more common from the 50th percentile onwards.This shows that there is a lack of redistributive gains from the minimumwage. While rich individuals finance the minimum wage raises on low wageformal workers, these mechanisms are operating within the richer householddeciles.562.8. Conclusions2.8 ConclusionsMinimum wages, while being a commonly used tool, remain a controver-sial policy. The two previous chapters argued that the presence of informalself-employment mutes the two main channels from which minimum wagescan benefit: efficiency and redistribution. In terms of the former channel,informal self-employment makes harder for firms to adjust in terms of wagesand prices, and thus inducing layoffs and transitions out of the formal sec-tor. Furthermore, despite the fact that mainly rich consumers enjoy thegoods and services produced and provided by minimum wage workers, theoverwhelming presence of informal workers in low-income households makesit hard for any sort of income redistribution towards them to exist.This chapter has focused on a redistribution analysis from a pecuniaryperspective. While these results point out that low-income individuals andhouseholds are certainly not harmed by the minimum wage increase, it couldbe that some workers are negatively impacted through other non-monetaryoutcomes. In Appendix A.3 I show, for instance, that minimum wage in-creases could perpetuate a cycle of being trapped in temporary jobs, buta more comprehensive analysis of these type of outcomes fall beyond thescope of this dissertation.One avenue of future research in this direction is to use the results of thispaper along with a theoretical model to optimally design minimum wagepolicies that are not set uniformly (i.e., at a national level), taking intoaccount the heterogeneous exposure to the informal sector across differentindustries or cities.572.9. Figures2.9 FiguresFigure 2.1: Consumption of goods produced using MW workersNote: This figure plots a binscatter of the share of total consumption produced by minimumwage workers - as described in Equation (2.3) - against the household monthly income in 2015.The red line corresponds to the best linear fit of that relationship. Source: ENAHO.582.9. FiguresFigure 2.2: Expected gains from MWNote: This figure plots the expected gains from the minimum wage as described in Equation(2.5). The elements of this Equation are jointly estimated and standard errors are bootstrapped.Source: PLAME and ENAHO.592.9.FiguresFigure 2.3: Effects on Household Income and Consumption(a) Household (HH) Income (b) Food (c) Rent and HH Maintenance(d) Furniture (e) Health (f) Transportation and Communica-tion(g) Leisure (h) MiscellaneousNote: This figure shows the effects of the minimum wage on changes in household income and consumption categories relative to householdconsumption in 2015. These estimates are computed based on Equation (2.2). Regressions are weighted by the logarithm of number of householdmembers in 2015, and robust standard errors are computed. Point estimates and 95% confidence intervals are shown. Source: ENAHO.602.9. FiguresFigure 2.4: Remaining in Informal SectorNote: This figure plots the minimum wage effects on remaining in the informal sector for workerswho were in that sector at baseline. The dependent variable is an indicator equal to 1 if the workerremains informal. The upper panel plots the coefficients from Equation (1.7) as of February 2015(pre) and February 2016 (post). The lower panel shows the difference between the blue and redline in the upper panel, which correspond to the coefficients of Equation (1.8). This regression isrun using the panel sample in the ENAHO dataset, which contains incumbent informal workers.Source: ENAHO.612.9. FiguresFigure 2.5: Effects on Income for Informal IncumbentsNote: This figure plots the minimum wage effects on remaining in the informal sector for workerswho were in that sector at baseline. The dependent variable is the change in income for workerswho were informal at baseline. The upper panel plots the coefficients from Equation (1.7) as ofFebruary 2015 (pre) and February 2016 (post). The lower panel shows the difference betweenthe blue and red line in the upper panel, which correspond to the coefficients of Equation (1.8).This regression is run using the panel sample in the ENAHO dataset, which contains incumbentinformal workers. Source: ENAHO.622.10. Tables2.10 Tables632.10.TablesTable 2.1: Incidence of the Minimum WageChanges Changesbetween 2015 between 2015and 2016 and 2017(1) (2)Change in total labor cost relative to revenue in 2015 0.0243 0.0327Change in revenue relative to revenue in 2015 (∆Revenue) 0.0757 0.0614Change in materials relative to revenue in 2015 (∆Material) 0.0065 -0.0028Change in miscitems relative to revenue in 2015 (∆MiscItems) 0.0453 0.0442Incidence on consumers (∆Revenue -∆Material -∆MiscItems) 0.239 0.02Change in profits relative to revenue in 2015 (∆Profit) -0.0007 -0.0095Change in depreciation relative to revenue in 2015 (∆Depr) 0.0004 -0.0032Incidence on firm owners (-∆Profit -∆Depr) 0.004 0.0127Fraction paid by consumers (percent) 98.35 61.12Fraction paid by firm owners (percent) 1.65 38.88Note: This table shows an accounting exercise to decompose the sources of changes in labor costs relative to revenuein 2015 (the year prior to the minimum wage hike). The first five rows compute the incidence on consumers andthe next three rows compute the incidence on firm owners. Finally, the last two rows divide the incidence withrespect to the change in labor costs to compute the fraction paid by consumers and firm owners, respectively.642.10.TablesTable 2.2: Do workers benefit from MW increase?Baseline bin t− 1 [650, 750) [750, 850) [950, 1050) [1050, 1150) [1150, 2050) [2050,max)Panel A. Income ChangeMean MW formal wage (w¯b) 746 818 903 1,001 1,479 2,630Effect exp. income (∆Ib/wb) 0.11 0.08 0.05 0.04 0.02 0.00∆ Ib/wb × w¯b 82.06 65.44 45.15 40.04 29.58 0.00Panel B. Expenses ChangeShare cons by MW workers (seMW ) 0.23 0.23 0.24 0.24 0.24 0.26Mean per cap. expenses (Eb) 590 643 700 613 792 1379∆ Eb (labor cost): 0.98× 0.123× seMWEb 16.35 17.82 20.25 17.73 22.91 43.22∆ Eb (manuf. price): 0.40× seMWEb 54.28 59.15 67.20 58.85 76.03 143.42∆ Eb (food price): 0.08× seMWEb 10.85 11.83 13.44 11.77 15.21 28.68Note: This table shows a redistribution analysis of the minimum wage at a worker-level. Columns indicate different earnings bins of formal workers,and minimum wage workers are those who earned below 850 PEN. Panel A computes the expected income gains - in levels - of formal workers basedon the reduced form estimates. Next, Panel B uses different approaches to estimate the expected change in expenses based on the reduced formestimates. I can thus compare, for each bin, changes in income versus changes in expenses.652.10.TablesTable 2.3: Do households benefit from MW increase?HH income decile 1 2 3 4 5 6 7 8 9 10Panel A. Income ChangeShare formal emp 0.01 0.03 0.11 0.19 0.26 0.34 0.46 0.60 0.72 0.82Share MW formal emp (slMW ) 0.00 0.02 0.05 0.1 0.13 0.15 0.19 0.21 0.22 0.15Mean MW formal income (I lMW ) 1 7 40 128 242 316 462 570 705 586∆IMWIMW× I lMW 0.00 0.42 2.40 7.68 14.52 18.96 27.72 34.20 42.30 35.16Panel B. Expenses ChangeShare cons by MW workers (seMW ) 0.16 0.19 0.2 0.21 0.22 0.22 0.23 0.23 0.24 0.27Mean expenses (E) 639 732 991 1,206 1,485 1,786 2,069 2,466 2,964 4,801∆ E (labor cost): 0.98× 0.123× seMWE 12.32 16.76 23.89 30.52 39.38 47.36 57.36 68.36 85.74 156.25∆ E (manuf. price): 0.40× seMWE 40.90 55.63 79.28 101.3 130.7 157.2 190.3 226.8 284.5 518.5∆ E (food price): 0.08× seMWE 8.18 11.13 15.85 20.26 26.14 31.43 38.07 45.37 56.91 103.7Note: This table shows the expected income changes versus expenses changes at the household level. It is divided by household income decilesacross columns. The first two rows in Panel A show the share of workers under formal employment and then restricting only to those who earnbelow 850 PEN (i.e. minimum wage workers), respectively.66Chapter 3Estimating the distributionof treatment effects indifference-in-differences: Adeconvolution approach3.1 IntroductionIn recent years the differences-in-differences research design has gained a lotof popularity due to increasing availability of high quality panel data in-formation. The target treatment effect parameter in this context is usuallythe Average Treatment Effect on the Treated (ATT) on different outcomes.Studying average effects in such fashion can exhibit some limitations. First,averages can mask substantial heterogeneity that is important to under-stand for policymaking. For instance, we can expect most policies to gen-erate some winners and losers and thus have different implications on theoutcome inequality. Secondly, this approach cannot analyze how treatmenteffects relate across outcomes to study the sources of the effect on the mainoutcome.It is well known that, even in the univariate case, the identification ofthe distribution of individual treatment effects poses some important chal-lenges. First, we do not simultaneously observe the treated and untreatedpotential outcomes for a given individual. This means that we must pro-vide assumptions to identify both marginal potential outcomes distributions.Secondly, even with information on the marginal distributions, identificationof the individual treatment effects require some assumptions about the de-pendence between potential outcomes. One approach is to rely on partialidentification methods (Heckman et al., 1997) to narrow down the class offunctions. Notably, Callaway (2021) has done significant progress, by invok-ing an assumption of stationarity of the dependence (or copula) of untreatedpotential outcomes over time, to establish tighter bounds on the distribu-673.1. Introductiontion of treatment effects. On the other hand, common approaches rely onvery strong restrictions of the dependence between treated and untreatedpotential outcomes (for example, the rank invariance assumption). Evenif the marginal distributions involving the treatment effects were observed,we still lack information about where treated individuals would be in the(counterfactual) distribution of non-treated outcomes.This paper proposes a more plausible set of assumptions to establishpoint-identification of the joint distribution of treatment effects. In partic-ular, we develop a deconvolution framework to identify such this treatmenteffect parameter in the case of longitudinal data with at least two timeperiods and a binary treatment.We show that point-identification is achieved via two key assumptions.The first assumption imposes that the untreated potential outcomes trendsare parallel across treatment status. This assumption is referred as StrongParallel Trends by Callaway and Li (2019). The second assumption statesthat the untreated potential outcomes trends are parallel across individualtreatment effects. Furthermore, this framework can easily accommodate theuse of covariates and multiple time-periods in staggered adoption settings(Callaway and Sant’Anna, 2020). Finally, we show the validity of the Gaus-sian multiplier bootstrap to construct uniform confidence bands, building onprevious work on intermediate Gaussian approximation by Chernozhukovet al. (2014, 2016).Related Literature. This paper contributes to a strand of literature thatfocuses on empirical evidence of treatment effects heterogeneity (Kottelen-berg and Lehrer, 2017; Djebbari and Smith, 2008), and the study of inequal-ity caused by policy shifts (DiNardo et al., 1996; Firpo et al., 2009; Atheyand Imbens, 2006; Melly and Santangelo, 2015)In a recent contribution, Callaway (2021) relies on a partial identifica-tion approach that bounds the joint distribution of treated and untreatedpotential outcomes under the assumption of constant dependence in thecopula function. In this paper, the author states the implausibility of thecurrent point-identification assumptions. We show a set of assumptions thatachieve this result without imposing the existing strong assumptions on thedependence between treated and untreated potential outcomes.This paper is also related to the use of deconvolution in measurementerror models. The use of deconvolution methods to achieve identificationof parameters of interest has been extensive as well. Notably, Bonhommeand Sauder (2012) focus on the identification of the marginal distribution of683.1. Introductionuntreated potential outcomes in the context of selective secondary educationand children’s test scores. Under some functional form assumptions, theyare able to identify such counterfactual distributions while allowing for (un-observed) children’s initial endowment to be correlated with the treatmentassignment. The main distinction with our study is that we focus on the dis-tribution of the individual treatment effects. Heckman et al. (1997) discussthe use of random coefficients model and deconvolution for the case of pro-gramme evaluations and cross-sectional data. They also discuss some mildtest to check whether identification of the distribution of treatment effectsis valid in their context. In a rather separate strand of literature, Arellanoand Bonhomme (2012) use the deconvolution approach to identify and es-timate distributional features of random coefficients in a linear panel dataframework, but target a different treatment effect parameter that requiresimposing additional assumptions on the error distribution. Additionally,Hall and Lahiri (2008); Dattner et al. (2011) study the estimation of dis-tribution of the unobserved signal in these type of models. I invite interestreaders to refer Schennach (2022) for a comprehensive literature review onthe use of these models in economics.Finally, this work is related to the problem of doing asymptotic inferencein deconvolution problems. This paper builds on results by Chernozhukovet al. (2016); Kato and Sasaki (2018); Adusumili et al. (2020); Bissantzet al. (2007) to establish the validity of uniform confidence bands for thejoint cumulative distribution function of treatment effects using a Gaussianmultiplier bootstrap.Outline. The rest of the paper is organized as follows. Section 3.2 presentsthe main framework of the paper and the identification results. Section 3.3proposes an estimator of the distribution of treatment effects, and discussesasymptotic theory and inference via multiplier bootstrap. Section 3.4 ex-tends our framework to the case of having access to covariates, and to amulti-period model. Finally, Section 3.5 concludes the paper. In Appendix3.6, we present proofs for the results and provide some additional details.Notation and convention. Let (Ω,A,P) be the underlying probabilityspace. For two random variables ξ and η, we write ξd= η if they havethe same distribution. For a random variable ξt, we denote one-perioddifferences as ∆ξt = ξt − ξt−1, and long differences g periods apart as∆gξt = ξt − ξt−g. We denote the characteristic function of random vari-able ξ as Ψξ(λ) ≡ E[exp(iλξ)], where λ ∈ R is the frequency and i ≡√−1is the imaginary unit.693.2. SetupFor any probability measure Q on a measurable space (S,S), we usethe notation Qf ≡ ∫ fdQ. Let Lp(Q), p ∈ [1,∞], denote the space of allmeasurable functions f : S → R such that ∥f∥Q,p ≡ (Q|f |p)1/p < ∞. Weuse the notation ∥f∥S ≡ supx∈S |f(x)|. Denote by eQ the L2(Q)-semimetricgiven by eQ(f, g) = ∥f − g∥Q,2, f, g ∈ L2(Q).For an arbitrary set T , let ℓ∞(T ) denote the space of all bounded func-tions T → R, equipped with the uniform norm ∥f∥T ≡ supt∈T |f(t)|. Wealso endow ℓ∞(T ) with the Borel σ−field induced from the norm topology.A random variable in ℓ∞(T ) refers to a Borel measurable map from Ω toℓ∞(T ). For ε > 0, an ε-net of a semimetric space (T, d) is a subset Tε of Tsuch that for every t ∈ T there exists a point tε ∈ Tε with d(t, tε) < ε. Theε-covering number N (T, d, ε) of T is the infimum of the cardinality of ε-nets.For a subset A of a semimetric space (T, d), let Aδ denote the δ-enlargementof A, that is, Aδ = {x ∈ T : d(x,A) ≤ δ}, where d(x,A) = infy∈A d(x, y).The standard Euclidean norm is denoted by |·|. For a, b ∈ R, we use thenotation a∨b = max{a, b}. Along with the imaginary unit and expectations,vectors are also denoted by boldface letters. The transpose of a vector xis denoted by x⊤. We write a ≲ b if there exists a universal constantC > 0 such that a ≤ Cb. Unless otherwise stated, c, C > 0 denote universalconstants of which the values might change from place to place. For anarbitrary function ν(t), we denote ∇tν(t) ≡ ν(t)− ν(−t).3.2 SetupWe first introduce our basic setup. We begin with a canonical model withtwo time periods t ∈ {0, 1}, two treatment groups d ∈ {0, 1} and two out-comes m ∈ {1, 2}. We define Di as an indicator that takes the value of 1when individual i belongs to treatment group d = 1. No unit is treated atthe initial time period t = 0, and units in the treatment group are impactedat t = 1.Next, we set up the potential outcomes framework. The observed po-tential outcomes for each unit i are given byYm,it = Ym,it(0) + (Ym,i(1)− Ym,i(0))︸ ︷︷ ︸τm,iDi 1{t = 1}, (3.1)where the term Ym,it(0) represents untreated potential outcomes of outcomem at time t, and τm,i represents the treatment effect at time t = 1 onoutcome m. We can take first-differences to the potential outcomes model703.2. Setupto obtain:∆Ym,i1 = ∆Ym,i1(0) + τm,i Di. (3.2)Our object of interest is the joint distribution function - across bothoutcomes m - of the treatment effects among the treated population (whosevalue of Di equals one), which are denoted by the vector τ i =[τ1i τ2i]⊤.The parameter of interest is given by the following quantityFτi|Di=1(r) = P (τ1i ≤ r1, τ2i ≤ r2 | Di = 1) .To keep the notation more concise, we will suppress the unit index iin our notation throughout the rest of the paper, except where it enhancesclarity. We make the following assumptions.Assumption ID. (i) The random vector ∆Y(0) is independent of D (ii)The random vector ∆Y(0) is independent of τ conditional on D = 1.Assumption CF. The characteristic function Ψ∆Y|D=0(λ1, λ2) is non-zerofor all λ ∈ R2.Assumption ID(i) is the Strong Parallel Trends assumption described inCallaway and Li (2019). This assumption generalizes the standard paral-lel trend assumption towards the entire distribution (i.e. the character-istic function of the outcome trends is independent with respect to thetreatment). Assumption ID(ii) states that the untreated potential out-come trends, provide no information about the treatment effects amongthe treated population. This assumption would be violated if, for exam-ple, individuals who gain the most from treatment are the ones whose out-come trends ∆Y(0) were more negative. This assumption does not restricterror dynamics , in contrast to Arellano and Bonhomme (2012), becauseAssumption ID(i) already provides identification of the untreated potentialoutcome’s marginal distribution.To further illustrate the previous assumptions, consider the case of in-voluntary job separations caused by mass layoffs and its impact on earnings(Schmieder et al., 2023). As it is standard in this literature, the researchdesign relies on matching workers displaced through a mass layoff with sim-ilar workers who do not experience a mass layoff. Assumption ID(i) requiresthat the distribution of earnings changes - had they not experienced a masslayoff - are the same among treated and control groups, which is reasonableunder this matching approach. On the other hand, Assumption ID(ii) re-quires that factors affecting earnings changes in the absence of treatment713.2. Setupnot depend on the individual earnings losses that stem from being affectedby a mass layoff. If workers who would’ve experienced large positive changesin productivity (in the absence of treatment) also experienced smaller jobdisplacement losses, then this assumption would be violated. In particular,this assumption states that if treated workers had not experienced a masslayoff, their idiosyncratic changes in earnings wouldn’t provide any infor-mation about who gains or loses more from experiencing a mass layoff18.This is why we interpret these idiosyncratic changes as the noise componentof a measurement error model relating changes in treated and untreatedpotential outcomes.Assumption CF states that the characteristic function of the untreatedpotential outcome trends are bounded away from zero. There are manyparametric distributions that satisfy this condition, and it is important tomention that this is an assumption that can be relaxed. More specifically,the sufficient condition is that the real zeros of the characteristic functionare isolated (see Evdokimov and White, 2012).The final identifying assumption we impose is that the treatment effectsare continuously distributed.Assumption DT. The random vector τ is continuously distributed.Assumption DT is a technical condition that allows to use known Fourierinversion formulas for continuous distributions (Gil-Pelaez, 1951; Shephard,1991).Theorem 1. Suppose that Assumptions ID, CF, and DT hold. Then, theunivariate and joint distribution satisfyFτm|D=1(r) =12− 1π∞∫0Im(e−iλmrλmΨ∆Ym|D=1(λm)Ψ∆Ym|D=0(λm))dλm , (3.3)− 2π2∞∫0∞∫0∇λ2Re(e−iλ⊤rλ1λ2Ψ∆Y|D=1(λ1, λ2)Ψ∆Y|D=0(λ1, λ2))dλ1dλ2= 4Fτ |D=1(r1, r2)− 2[Fτ1|D=1(r1) + Fτ2|D=1(r2)]+ 1 . (3.4)Theorem 1 is the main identification result of this paper. The proofstates that under the previous assumptions we can identify the characteristic18In reality, this assumption needs to hold after controlling for observables. This relaxingassumption is discussed in Section 3.4.723.3. Estimation and Asymptotic Inferencefunction of τ among the treated group, and the result thus follows from theFourier inversion formula.3.3 Estimation and Asymptotic InferenceSuppose we observe an independent sample {Y1,i0, Y2,i0, Y1,i1, Y2,i1, Di}ni=1.In this section we will describe how to perform the estimation of the dis-tribution function and uniform confidence bands over a compact intervalr ∈ I. The result builds on an intermediate Gaussian approximation (Katoand Sasaki, 2018; Chernozhukov et al., 2016).3.3.1 EstimationWe will assume for simplicity that E[D] is known, given that its estimatorconverges at a parametric rate and is thus asymptotically negligible in theupcoming discussions. We can estimate the characteristic functions asΨˆ∆Y|D=1(λ) =1E[D]1nn∑i=1exp(iλ⊤∆Yi)Di , andΨˆ∆Y|D=0(λ) =11−E[D]1nn∑i=1exp(iλ⊤∆Yi)(1−Di).To perform estimation and inference for the cumulative distribution func-tion we define an infeasible estimator in which we assume we know thecharacteristic function in the denominatorFˆ ∗τm|D=1(r) =12− 1πh−1∫hIm(e−jλmrλmΨˆ∆Ym|D=1(λm)Ψ∆Ym|D=0(λm))ΨK(hλm)dλm,− 2π2h−1∫hh−1∫h∇λ2Re(e−iλ⊤rλ1λ2Ψˆ∆Y|D=1(λ1, λ2)Ψ∆Y|D=0(λ1, λ2))ΨK(hλ1)ΨK(hλ2)dλ1dλ2= 4Fˆ ∗τ1,τ2|D=1(r1, r2)− 2[Fˆ ∗τ1|D=1(r1) + Fˆ∗τ2|D=1(r2)]+ 1, (3.5)which can also be written asF ∗τm|D=1(r) =12− 1nn∑i=1Lm,n(r −∆Ym,ih),733.3. Estimation and Asymptotic InferenceF ∗τ |D=1(r) =12[F ∗τ1|D=1(r1) + F∗τ2|D=1(r2)]− 14(3.6)− 1nn∑i=1Kn(r−∆Yih),where Lm,n for m ∈ {1, 2} and Kn are defined as follows:Lm,n(r) =1π1∫h21λIm[e−iλrΨ∆Ym|D=0(λ/h)]ΨK(λ)dλ,Kn(r) =12π21∫h21∫h2∇λ2Re(e−iλ⊤rλ1λ21Ψ∆Y|D=0(λ/h))ΨK(λ1)ΨK(λ2)dλ1dλ2.The kernel function K : R→ R is real-valued and such that the Fouriertransform ΨK is supported in the interval [−1, 1]. The bandwidth satisfiesh → 0 as n → ∞. The functions Lm,n and Kn are called the deconvolutionkernels.3.3.2 Asymptotic InferenceWe will show that one can construct confidence bands Cn of level approx-imately 1 − α (as n grows to infinity) for Fτ |D=1. First, we will describeinformally how the methodology works. Next, we state the assumptions andmain results.MethodologyThe idea behind the approach is to establish conditions such that the esti-mation noise of the numerator terms of the estimators in (3.5) dominate theone in the denominator term. First, we establish a Gaussian approximationof the supremum in absolute value of the following stochastic process:Z∗n(r) =Fˆ ∗τ |D=1(r)−E[Fˆ ∗τ |D=1(r)]√Var(Fˆ ∗τ |D=1(r))(3.7)=√n{Fˆ ∗τ |D=1(r)−E[Fˆ ∗τ |D=1(r)]}σn(r), r ∈ I,where σ2n(r) ≡ Var(∑2m=1 Lm,n(rm−∆Ymh)+Kn(r1−∆Y1h ,r2−∆Y2h))> 0 forall r ∈ I.743.3. Estimation and Asymptotic InferenceBy doing so we are characterizing the stochastic part while not account-ing for the bias E[Fˆ ∗τ |D=1(r)]− Fτ |D=1(r). Therefore, the upcoming re-sults will be used to derive confidence bands for E[Fˆ ∗τ |D=1(r)]initially. InCorollary 1 coming up ahead, we layout the conditions that need to hold forthese confidence bands to also be valid for the true cumulative distributionfunction. The result will require undersmoothing the bandwidth.In practice, it is impossible to construct such a confidence band becausethe distribution of ||Z∗n||I and the variance function σ2n(·) are unknown.Therefore, we will have to construct feasible version by using the sampleanalogues, Lˆm,n and Kˆn, of the deconvolution kernels.Similarly, we can also construct a feasible estimator of the variance func-tion σ2n(r) in the following way:σˆ2n(r) =1nn∑i=1[2∑m=1Lˆm,n(rm −∆Ym,ih)+ Kˆn(r−∆Yih)]2−{1nn∑i=1[2∑m=1Lˆm,n(rm −∆Ym,ih)+ Kˆn(r−∆Yih)]}2(3.8)Next, consider the following Studentized processZˆn(r) =√n{Fˆτ |D=1(r)−E[Fˆ ∗τ |D=1(r)]}σˆn(r), r ∈ I, (3.9)The quantiles cn(1−α) of the Studentized process are still unknown, butwe will show that under our assumptions (an important piece of it is howto control the rate of the variance function), the distribution of ||Zˆn||I canbe approximated by the supremum of the absolute value of a tight Gaussianrandom variable ZGn in ℓ∞(I) with mean zero and the same covariance asZ∗n. This motivates a Gaussian multiplier bootstrap approach as in Katoand Sasaki (2018) and built on the work of Chernozhukov et al. (2016).To implement this approach, we generate independent standard normalrandom variables ξ1, . . . , ξn ∼ N(0, 1), independently of the data Dn =753.3. Estimation and Asymptotic Inference{Y1,i0, Y2,i0, Y1,i1, Y2,i1, Di}ni=1, and construct the multiplier processZˆξn(r) =1σˆn(r)√nn∑i=1ξi{Lˆ1,n(r1 −∆Y1,ih)− 1nn∑l=1Lˆ1,n(r1 −∆Y1,lh)+ Lˆ2,n(r2 −∆Y2,ih)− 1nn∑l=1Lˆ2,n(r2 −∆Y2,lh)(3.10)+Kˆn(r−∆Yih)− 1nn∑l=1Kˆn(r−∆Ylh)},for r ∈ I. Conditionally on the data, the multiplier process is a Gaussianprocess with mean zero and the covariance function sufficiently close to thatof ZGn .In particular, we show that we can estimate the quantiles cn(1 − α) of||Zˆn||I by computing the (1− α)-quantile of ||Zˆξn||I conditional on the dataDn. Finally, the resulting confidence band is given byCˆn(r) =[Fˆτ (r)± σˆn√ncˆn(1− α)], r ∈ I, r ∈ I.Next, we will formally state the necessary assumptions and main resultsin terms of inference.Assumptions and Main Results: InferenceIn this section we introduce the two main theorems that ensure theasymptotic validity of the confidence bands constructed using the multiplierbootstrap approach. We begin by describing all the necessary assumptions.Assumption C. (i) {Y1,i0, Y2,i0, Y1,i1, Y2,i1, Di}ni=1 is an i.i.d. sample fromthe potential outcomes model in (3.1).(ii) The joint densities f∆Y and fτ |D=1are bounded and continuous on R2, and infr∈I f∆Y(r) > c for some c > 0and compact set I. (iii) The characteristic function of τ givenD = 1 satisfiessupω∈R2[2∏m=1(1 + |ωm|)γ |Ψτ |D=1(ω)|]< Cfor some γ,C > 0.Assumption C(i) implies that the model in (3.2), conditional on thetreated group, is in essence a measurement error model with unknown errordistribution ∆Y(0). However, given our identification assumptions, we have763.3. Estimation and Asymptotic Inferenceaccess to a sample (control group) drawn from this unknown distribution.Assumption C(ii) is a mild condition but can exclude some error distribu-tions such as the Cauchy distribution. Assumption C(iii) is a Sobolev con-dition, which characterizes the smoothness of the density of τ given D = 1,and is used to bound the rate of the bias term.Assumption OS. (i) There exists some αu > 1/2 and c1, C1, λ0 > 0 suchthatc12∏m=1|λm|−αu1{|λm|≥λ0} ≤ |Ψ∆Y|D=0(λ)| ≤ C12∏m=1|λm|−αu1{|λm|≥λ0},for all λ such that at least one λm satisfies |λm| ≥ λ0 (ii) K is an evenfunction with ΨK(λ) = (1− λq)p1{|λ| ≤ 1} for some q, p ≥ 2.Assumption OS(i) is a condition that characterizes ordinary smooth den-sities. The case where αu > 1/2 exclude cases where the cumulative distri-bution function estimator Fˆτ |D=1 converges at a parametric rate (Dattneret al., 2011). Assumption OS(ii) imposes conditions on the kernel func-tion. In particular, we impose a common choice for K and ensure that thedeconvolution kernels are L1-integrable.The next assumption restricts the rate at which the bandwidth h can goto zero.Assumption BW. The bandwidth satisfies1. logn log2/3 h−1n1/6h2/3√log h−1 → 0.2. n−1/2h−2−2αu√log h−1 → 0.3. n−1/2h−2 log5 h−1∨n−1/2h−2−2αu log7/2 h−1 → 04. n−1/2h−4+2αu log h−1 → 0.The first term of Assumption BW ensures the intermediate Gaussian ap-proximation of the infeasible process Z∗n(r) (Chernozhukov et al., 2014). Thesecond term controls the estimation error due to the characteristic functionin the denominator. The other two terms are used to control the estimationerror of the variance term.Theorem 2. Suppose that Assumptions C, OS and BW hold. Recall theStudentized process Zˆn defined in Equation (3.9). For each sufficiently largen, there exists a tight Gaussian random variable ZGn ∈ ℓ∞(I) with mean zero773.3. Estimation and Asymptotic Inferenceand the same covariance function as Z∗n defined in Equation (3.7), such thatas n→∞,supz∈R2|P{∥Zˆn∥I ≤ z}− P {∥ZGn ∥I ≤ z}| → 0,Theorem 2 ensures that the supremum over I of the Studentized processZˆn can be approximated by an intermediate Gaussian random variable. Asit was previously discussed, we cannot perform inference with this resultyet, as it contains unknown components for its computation.Next, we show the validity of the Gaussian multiplier bootstrap in thissetting.Theorem 3. Suppose that Assumptions C, OS and BW hold. Recall themultiplier processes Zˆξn defined in Equation (3.17). Then, as n→∞,supz∈R2|P{∥Zˆξn∥I ≤ z | Dn}− P {∥ZGn ∥I ≤ z}| →p 0, (3.11)where Dn = {Y1,i0, Y2,i0, Y1,i1, Y2,i1, Di}ni=1, and ZGn is the Gaussian randomvariable in ℓ∞(I) given in Theorem 2. Therefore, letting cˆn(1 − α) denotethe (1 − α)-quantile of the conditional distribution of ∥Zˆξn∥I given Dn, wehave thatP{∥Zˆn∥I ≤ cˆn(1− α)}→ 1− α (3.12)as n → ∞. Finally, the supremum width of the confidence band Cˆn isOp(√log h−1h−2α+2√n).Theorem 3 indicates that by constructing the multiplier process, we canuse the quantiles of the supremum (in absolute values) over r ∈ I to con-struct asymptotically valid uniform confidence bands of the joint distributionof treatment effects.Remark 1. The Studentized process considers only the variance part. Todeal with the bias part we are required to undersmooth the bandwidth. Weintroduce next how to extend the main results to construct a valid confidenceset for Fτ instead of E[F∗τ ].Corollary 1. To control the bias we further assume thatn1/2h2(αu+γ−1)√log h−1 → 0,783.4. Extensionswhere γ is the parameter in the Sobolev condition Assumption C(iii). Then,as n grows to ∞, the multiplier bootstrap confidence band Cˆn satisfiesP(Fτ (r) ∈ Cˆn ∀r ∈ I)→ 1− α.Remark 2. In this paper we are focused on an ordinary smooth error dis-tribution (see Assumption OS(i)), in which the decay of the characteristicfunction as the magnitude of λ goes to infinity has a polynomial rate. Thecase of exponential decay is known as supersmooth error distributions. Weexclude this case, as it is known that the estimation noise in the denomi-nator of Ψτ |D=1 will dominate the one in the numerator, thus invalidatingthe use of a multiplier bootstrap approach (Adusumili et al., 2020; Kurisuand Otsu, 2022). This setting increases the difficulty of any deconvolutionapproach significantly, and thus require more careful construction of boot-strap approach (e.g., estimating the numerator using the full sample and thedenominator using a subsample across bootstrap repetitions to control whichestimation noise dominates).3.4 ExtensionsThis section expands the identification arguments from the basic setup toallow for (i) adding covariates to the model, (ii) multiple periods in a stag-gered adoption setting, and (iii) analyzing multiple (more than 2) outcomevariables.3.4.1 Adding CovariatesTo accommodate covariates in this framework, we propose partialling outthe effect of observables. In particular, we can augment the structure aroundthe potential outcomes model described in Section 3.1Ym,t(d) = τm1{t = 1}1{d = 1}+X⊤mγm,t + νm + εm,t.By taking first differences to the previous model we obtain the followingexpression:∆Ym,1 = τmDi +X⊤m∆γm,1 +∆εm,1.The objective is then to estimate ∆γm,1 across different outcomes. Todo so, we use the subset of individuals in the control group and impose thefollowing assumption793.4. ExtensionsAssumption X. The random variables ∆εt and Xm satisfyE [∆εt | Xm, D = 0] = 0, and E[X⊤mXm | D = 0]is positive definite.Given this result, identification of ∆γm,1 follows and is given by:∆γm,1 = E[Xm,iX⊤m,i | D = 0]−1E [Xm,i∆Ym,i1 | D = 0] ,which can then be estimated by its sample analogue. We then propose topartial out the outcome to obtain∼∆Y m ≡ ∆Ym,1 −X⊤m∆̂γm,1, and proceedas in the case of not having covariates in the model.Remark 3. It is also possible to state Assumption ID conditional on observ-ables X, and compute conditional expectations non-parametrically through-out the estimation process (e.g., Nadaraya-Watson estimator). We do notfollow this path as limited overlap can generate computational issues withour deconvolution method, as well as imprecise inference.3.4.2 Staggered AdoptionIn this subsection we show how to extend the basic setup to a multi-periodmodel with staggered adoption. The model now consists of t = 1, . . . , Tperiods. Let Dt be an indicator variable equal to one if the unit is treatedat period t and zero otherwise. We make the following assumption aboutthe treatment process:Assumption T-SA. D1 = 0 a.s. for all units. Moreover, for t = 2, . . . , T ,Dt−1 ≥ Dt almost surely.Assumption Assumption T-SA states that no unit is treated at t = 1;and once treatment is activated for some unit, it can’t be turned off in thefuture. This assumption is also known as staggered treatment adoption.We define Gi as the time period when some unit i is first treated. Forinstance, if unit i enters treatment at t = 3, then G = 3 for that unit. Fornever-treated units we set G = ∞. Define Gg = 1{G = g} = 1{Dg = 1};in other words, it is equal to one when unit i is first treated in period g.Similarly, we define C as an indicator that is equal to one when unit i isnever treated (i.e., G = ∞). Suppose we pool the never-treated group andsome group of individuals that enter treatment at t = g.For t = 1, . . . , T , let Yit(g) denote the potential outcome that unit iwould experience at time t if they were to first become treated in time803.4. Extensionsperiod d. The observed and potential outcome m for each unit i are relatedthrough the following equation:Ym,t = Ym,t(0) +∑g∈G(Yt(g)− Yt(0))︸ ︷︷ ︸τm(t,g)Gg (3.13)This equation describes the fact that we only observe one potential out-come path for each unit. Observed outcomes are untreated potential out-comes in all periods for the never-treated group. For units that do par-ticipate in the treatment, observed outcomes are the unit-specific potentialoutcomes corresponding to the particular time period when that unit adoptsthe treatment.We denote τm(t, g) ≡ Ym,t(g)− Ym,t(0) the treatment effect for outcomem at time t > g, had they been first treated at time period g (in otherwords, it also varies according to the treatment cohort). These are thegeneralization of the treatment effect parameters referred as ATT(t, g) byCallaway and Sant’Anna (2020).Assumption ID-SA. The following conditions hold for all t = 2, . . . , Tand g = 2, . . . , T such that t > g(i) ∆gYt(g) is independent of Gg conditional on Gg + C = 1.(ii) ∆gYt(g) is independent of τ (t, g) conditional on Gg = 1.Assumption CF-SA. The characteristic function Ψ∆gY|C=1(λ) is non-zerofor all λ ∈ R2.The last assumption is that the parameters τ (t, g) have a continuousdistribution, so that we are able to invert their characteristic function toobtain the cumulative distribution function.Assumption DT-SA. τ (t, g) are continuously distributed.The parameters of interest in this framework are the distribution func-tions of τ (t, g) . The following theorem extends the results from Section 3.2towards this multiple-period model.Theorem 4. Suppose Assumptions ID-SA, CF-SA and DT-SA hold. Then,for all t > g, the univariate and joint distribution satisfyFτm(t,g)|Gg=1(r) =12− 1π∞∫0Ime−iλmrλmΨ∆gYm|Gg=1(λm)Ψ∆gYm|C=1(λm) dλm (3.14)813.4. Extensions− 2π2∞∫0∞∫0∇λ2Ree−iλ⊤rλ1λ2Ψ∆gY|Gg=1(λ1, λ2)Ψ∆gY|C=1(λ1, λ2) dλ1dλ2= 4Fτ (t,g)|Gg=1(r1, r2)− 2[Fτ1(t,g)|Gg=1(r1) + Fτ2(t,g)|Gg=1(r2)]+ 1 (3.15)Theorem 4 states that, with access to a never-treated group, we can nat-urally extend the framework in Section 3.2 towards a setting with multipleperiods and staggered adoption. In particular, to identify the parameterτ (t, g) for a particular time period t and cohort g, we only require to do thefollowing procedure1. Subset the data to only contain Gg = 1 and C = 1.2. Take a long difference between time t and g.3. Obtain the joint cumulative distribution of τ (t, g) as if we were in thebasic setup (i.e., with two time periods).In other words, once we perform these transformations we are back to thebasic model. An implication of this is that we can apply the same estimationand inference tools discussed in Section 3.3.Remark 4. We can easily accomodate control variables in this setting aswell. In particular, suppose that the observed outcomes satisfyYm,t = X⊤mγm,t +T∑g=2τm(t, g)Gg + v + εt. (3.16)By taking differences between time periods t > g and g, the parameter ∆gγm,tcan be identified by analyzing individuals in the control group and invokingthe following assumption akin to the one discussed at the beginning of thissection:Assumption X-SA. For all t = 2, . . . , T and g = 2, . . . , T such that t > g,E [∆gεt | Xm, C = 1] = 0, and E[X ′mXm | C = 1]is positive definite.Hence, we can partial out the outcome variable and proceed as if we werein a no covariates model.823.4. ExtensionsRemark 5. In this setting it is also possible to check for pre-trends bycomputing τ (t, g) for t < g. A large probability mass away from zero wouldindicate that the outcome trends between the treated and untreated groupsdiffer systematically in time periods t when the cohort group g has not yetbeen treated.3.4.3 Beyond two outcomesIn this section we will return to the basic model with two time periods andtwo treatment groups. Given that the identification of the characteristicfunction of τ among the treated holds for any arbitrary number of outcomesM (see the proof of Lemma 2), it suffices to show a Fourier inversion formularesult that holds in higher dimensions.The next result establishes the key result in Shephard (1991) in thegeneral case of many outcomes.Theorem 5 (Shephard, 1991). Suppose that Assumptions ID, CF, andDT hold for the case of M different outcomes. Then, if fτ |D=1,Ψτ |D=1 ∈LM (P ), the following equality holds(−2)M(2π)M∞∫0· · ·∞∫0∇λ1. . . ∇λM[e−ir⊤λiλ1iλ2 . . . iλpΨ∆Ym|D=1(λ)Ψ∆Ym|D=0(λ)]dλ = ν†(r),whereν†(r) = 2MFτ |D=1(r1, . . . , rM )− 2M−2 [Fτ |D=1(r2, r3, . . . , rM ) + · · ·+ Fτ |D=1(r1, . . . , rM−2, rM−1)]+ 2M−2[Fτ |D=1(r3, r4, . . . , rM ) + · · ·+ Fτ |D=1(r1, . . . , rM−3, rM−2)]+ · · ·+ (−1)M .The importance of Theorem 5 is that it shows that we can successivelygenerate the distribution functions from cumulative distribution functionsin a sequential way. For example, we begin by computing univariate cumu-lative distribution functions Fτ |D=1(r1), then compute bivariate distributionfunctions Fτ |D=1(r1, r2), and so forth.The following corollary leverages some trigonometric identities to obtaina different representation depending on whether the dimensionM is even orodd.833.5. ConclusionsCorollary 2 (Shephard, 1991). Under the existence of the mean of τ givenD = 1, the following equality holds.∇λ1. . .∇λp[e−ir⊤λiλ1iλ2 . . . iλMΨ∆Y|D=1(λ)Ψ∆Y|D=0(λ)]= 2iM−1∇λ2. . . ∇λMIm[e−ir⊤λiλ1iλ2 . . . iλpΨ∆Y|D=1(λ)Ψ∆Y|D=0(λ)],if M is odd= 2iM−1∇λ2. . . ∇λMRe[e−ir⊤λiλ1iλ2 . . . iλMΨ∆Y|D=1(λ)Ψ∆Y|D=0(λ)],if M is evenRemark 6. Estimation and asymptotic inference follow from this generalidentification result. An additional derivation of the rate of convergence inthis general case will be omitted. However, as the number of outcomes Mscales up rate of convergence will be slower through an adjustment we haveto make on the bandwidth.3.5 ConclusionsThis chapter has developed a new approach to point-identify and estimatea key parameter of interest to unmask unobserved treatment heterogeneity:the cumulative distribution of treatment effects across multiple outcomes.The key results depend on (i) access to at least two time periods of paneldata, (ii) distributional parallel trends assumption across treatment status- which ensures the identification of the marginal distribution of the trendof untreated potential outcomes among the treated - and (iii) a new paralleltrends assumption across individual-treatment effects among the treated,which states that the untreated trends do not provide information aboutindividual treatment effects within this group. Additionally, the frameworkcan easily incorporate (i) the use of covariates, (ii) multi-period settings withstaggered adoption, and (iii) an arbitrary number of outcomes - providedthese do not increase with sample size.We view this deconvolution approach as a flexible tool to obtain policy-relevant information from certain reforms or policies (e.g., study whetherpolicies increase or decrease inequality among the treated group), withoutthe drawback of having to rely on partial identification approaches that maynot be too informative.843.6. Proofs3.6 Proofs3.6.1 LemmasThis section discusses the Lemmas used in the proof of the main results ofthis chapter.Lemma 1 (Theorem 2.7.17 in Van der Vaart and Wellner (2023)). Supposethat we have a class of functions x 7→ ft(x) that is Lipschitz in the indexparameter t ∈ T . Suppose that|fs(x)− ft(x)| ≤ d(s, t)F (x),for some F (x) and distance d(s, t).Then F = {ft : t ∈ T } satisfies the following property for any norm ∥·∥,N[] (F , ∥·∥, 2ϵ∥F∥) ≤ N (T, d, ϵ),where N[] denotes the bracketing number and N the covering number.Lemma 2 (Identification of characteristic function). Let Assumptions IDand CF hold. ThenΨτ (λ) =Ψ∆Y|D=1(λ)Ψ∆Y|D=0(λ).Proof of Lemma 2 Rewrite the potential outcomes model as(∆Y1,i1∆Y2,i1)︸ ︷︷ ︸∆Y=(τ1,iτ2,i)︸ ︷︷ ︸τDi +(∆Y1,i1(0)∆Y2,i1(0))︸ ︷︷ ︸∆Y(0)Then, by computing the characteristic function given Di = 1 on bothsides we obtainE(exp(iλ⊤∆Y) | D = 1) = E(exp(iλ⊤τ + iλ⊤∆Y(0)) | D = 1)= E(exp(iλ⊤τ) | D = 1)E(exp(iλ⊤∆Y(0)) | D = 1)= E(exp(iλ⊤τ) | D = 1)E(exp(iλ⊤∆Y) | D = 0),where the second and third equality follow from Assumption ID(ii) andAssumption ID(i), respectively. Next, Assumption CF states that the lastterm on the RHS is bounded away from zero. Therefore, the desired resultfollows. ■853.6. ProofsLemma 3 (Adaptation of Theorem 2.1 in Chernozhukov et al. (2016)).Let F be a class of measurable functions on a measurable space (S,S) withenvelope F such that supQN (F , eQ, ε∥·∥Q,2) ≤ (A/ε)v for some A ≥ e,v ≥ 1, and all 0 < ε ≤ 1, where the supremum is taken over all finitelydiscrete probability measures on (S,S). In addition, assume(A) There exists constants b ≥ σ > 0 and q ∈ [4,∞) such that supf∈FP |f |k ≤σ2bk−2 for k = 2, 3, 4, and ∥F∥P,q ≤ b.(B) Let Kn = Kn(v,A, b, σ) = v (log n ∨ log(Ab/σ)) be such that K3n ≤ n.Let Z = supf∈F Gnf be the empirical process. Then for every γ ∈ (0, 1),there exists a random variable Z˜d= GP f , where GP is a centered Gaus-sian process indexed by F with uniformly eP -continuous sample paths andcovariance functionE[GP (f)GP (g)] = Cov (f(X), g(X)) , f, g ∈ F .Furthermore, the random variable Z˜ satisfiesP{|Z − Z˜| > C1δ(1)n } ≤ C2(γ + n−1),where C1, C2 are positive constants that depend only on q, andδ(1)n = δ(1)n (v,A, b, σ, q, γ) =bKnγ1/qn1/2−1/q+(bσ2K2n)1/3γ1/3n1/6.Lemma 4 (Uniform convergence of multivariate characteristic function).Let Ψ(λ) =∫eiλ⊤xdF (x) and let X1, . . . , Xn ∈ RM be an independent sam-ple from F , where M does not increase with n. Let Fn be the empiricaldistribution function and Ψn(λ) =1n∑ni=1 eiλ⊤Xi be the empirical charac-teristic function.Then||Ψn −Ψ||[−h−1,h−1] = sup|λm|≤h−1,∀m∈{1,··· ,M}|Ψn(λ)−Ψ(λ)|= Op{n−1/2√log h−1}.Proof of Lemma 4:Define the classGh ={gλ(·) : u 7→ eiλ⊤u, λ such that |λm| ≤ h−1, ∀m}.863.6. ProofsNext, notice that√n[Ψn(λ) − Ψ(λ)] = Gn(f), f ∈ Gh, where Gn(·) isthe empirical process given byGn(f) =1√nn∑i=1{f(Ui)−E[f(Ui)]}For any gλ1(·) , gλ2(·) ∈ Gh, with λ1,λ2 such that each of their compo-nents are within [−h−1, h−1], the Mean Value Theorem implies the following:|gλ1(·)− gλ2(·)| = |eiλ⊤1 u − eiλ⊤2 u|≲ ∥λ1 − λ2∥.Therefore, by Lemma 1 ,N (Gh, ∥·∥, ϵ∥F∥) ≤ N[](Gh, ∥·∥, 2ϵ∥F∥)≤ N ([−h−1, h−1]M , ∥·∥, 2ϵ)≤(Ah−1ϵ)M,for any norm ∥·∥, and A > e. Furthermore, the complex exponential isuniformly bounded by one, so that Gh = 1 is a valid envelope function forthis class. This implies thatsupQN (Gh, || · ||Q,2, ϵ||Gh||Q,2) ≤(Ah−1n /ϵ)M, 0 < ∀ϵ ≤ 1,where supQ is taken over all finitely discrete distributions on RM .Next, Lemma 3 with parameters Kn ∼ log n ∨ log h−1 = log n , b ∼ 1 ,q = ∞ and γ = 1/ log n ensures the existence of a tight Gaussian randomvariable Gn in ℓ∞(Gn) with mean zero and the same covariance functionas Gn. Furthermore, there exists a sequence of random variables Wn withWnd= supf∈Gh |Gn(f)| and such that∣∣∣ supf∈Gh|Gn(f)| −Wn∣∣∣ = Op( log nn1/2+log nn1/6)= op(1).By Corollary 2.2.9 in Van der Vaart and Wellner (2023), we obtainE[supf∈Gh|Gn(f)|]≲∫ 10√1 + log supQN (Gh, ∥·∥Q,2, ϵ)dϵ≲√log h−1.873.6. ProofsFinally, we have thatsup|λm|≤h−1,∀m∈{1,··· ,M}|Ψn(λ)−Ψ(λ)| = n−1/2 supf∈Gh|Gn(f)|= Op(n−1/2√log h−1),thus completing the proof. ■Lemma 5. Suppose Assumption OS(i) holds. Let λ ∈ RM , theninf|λm|≤h−1,∀m∈{1,··· ,M}|Ψ∆Y|D=0(λ)| ≥ c1 (1− o(1))hMαu .Proof of Lemma 5:Without loss of generality, we can consider the case where M = 1, asit eases the exposition. For values |λ| ≥ h−1 we have the lower boundgiven by Assumption OS(i). As n grows large, h−1 goes to infinity and thecharacteristic function decays to zero by the Riemann-Lebesgue Theorem.However, for fixed n the function |Ψ∆Y |D=0(λ)| oscillates between some c > 0and 1 among the set of λ ∈ [0, h−1] (the peak is attained at λ = 0, as it is acharacteristic function).Therefore, the best estimate for inf |λ|≤h−1 |Ψ∆Y |D=0(λ)| is given by thelower bound in Assumption OS(i) evaluated at h−1 minus some arbitrarilysmall number ϵn (due to the oscillating nature of the function). As n→∞the function decays to zero and thus we can take ϵn → 0. ■Lemma 6 (Variance lower bound). Recall σn as the variance function de-fined in Equation (3.7). Let let ΨK,H(λ) ≡ ΨK(λ1h)ΨK(λ2h), U1 ≡ ∆YDand U0 ≡ ∆Y(1−D), as it will simplify many algebraic expressions. Now,suppose Assumption OS hold, and assume that for any ϵ > 0 there exists anCϵ > 0 such that∫|u1|<hϵ∫|u2|<hϵ −12π21∫h21∫h2{sin(−λ1u1) sin(−λ2u2)−E(sin(−λ1U1,1h)sin(−λ2U1,2h))}×ΨK(λ1)ΨK(λ2)dλ1dλ2+hαU2∑m=11π1∫h2{sin(−λmum)−E(sin(−λmUmh))}ΨK(λmh)dλj2du1du2 > Cϵ,uniformly over n.Then for a compact interval I ⊂ R2 and n large enough, we have thatinfr∈Iσ2n(r) ≿ h−2(2αu−2).883.6. ProofsProof of Lemma 6:Writeσ2n(r) = E 12π2h−1∫hh−1∫hRe{e−iλ⊤rλ1λ2ΨˆU1 (λ)−ΨU1 (λ)ΨU0 (λ)}ΨK,H(λ)dλ1dλ2− 12π2h−1∫hh−1∫hRe{e−i(λ1r1−λ2r2)λ1λ2ΨˆU1 (λ1,−λ2)−ΨU1 (λ1,−λ2)ΨU0 (λ1,−λ2)}ΨK,H(λ)dλ1dλ2+1π2∑m=1h−1∫hIm{e−i(λmrm)λmΨˆU1,m (λm)−ΨU1,m (λm)ΨU0,m (λm)}ΨK(λmh)dλm2≿ h4−4αu∫R∫R 12π21∫h21∫h2{cos(−λ1u1 − λ2u2)−E(cos(−λ1U1,1 − λ2U1,2h))}×ΨK,H(λ)dλ1dλ2− 12π21∫h21∫h2{cos(−λ1u1 + λ2u2)−E(cos(−λ1U1,1 + λ2U1,2h))}ΨK,H(λ)dλ1dλ2+hαu2∑m=11π1∫h2{sin(−λmum)−E(sin(−λmUmh))}ΨK(λmh)dλm2× fU (r1 − hu1, r2 − hu2) du1du2≿ h4−4αu∫R∫R −12π21∫h21∫h2{sin(−λ1u1) sin(−λ2u2)−E(sin(−λ1U1,1h)sin(−λ2U1,2h))}×ΨK,H(λ)dλ1dλ2+hαu2∑m=11π1∫h2{sin(−λmum)−E(sin(−λmU1,mh))}ΨK(λmh)dλm2du1du2,where the second inequality comes from change of variables and As-sumption OS. The last inequality comes from infr∈I f∆Y(r) > 0 and thetrigonometric equality cos(a+ b) = cos(a) cos(b)− sin(a) sin(b). ■Lemma 7. The population and empirical characteristic functions satisfy893.6. Proofsthe following propertyE h−1∫λ0· · ·h−1∫λ0(M∏m=1λ−am)∣∣∣Ψˆn(λ)−Ψn(λ)∣∣∣2 dλ1 · · · dλM=n−1h−M(1−a) , if a < 1n−1 , if a > 1n−1M log h−1 , if a = 1Proof of Lemma 7:By Fubini’s theorem we can writeE h−1∫λ0· · ·h−1∫λ0(M∏m=1λ−aj)∣∣∣Ψˆn(λ)−Ψn(λ)∣∣∣2 dλ ≤ h−1∫λ0· · ·h−1∫λ0(M∏m=1λ−aj)×E[∣∣∣Ψˆn(λ)−Ψn(λ)∣∣∣2] dλ1 · · · dλM= O(n−1)M∏m=1(∫ hλ0λ−aj dλm),where the result follows from solving the integral. ■903.6. Proofs3.6.2 Proof of Theorem 1By applying Lemma 2 we identify the characteristic function of τ among thetreated group. The inversion of the cumulative distribution function followsfrom Assumption DT, along with the results in Shephard (1991) and Gil-Pelaez (1951) for the joint and univariate cumulative distribution functions,respectively. ■913.6. Proofs3.6.3 Proof of Theorem 2This proof will proceed in three steps, in the same spirit as Kato and Sasaki(2018). To simplify notation let ΨK,H(λ) ≡ ΨK(λ1h)ΨK(λ2h).First, we focus on the Gaussian approximation of the process Z∗n(r), de-fined in Equation (3.7), to a tight Gaussian in ℓ∞(I). Next, we will showthat the estimation noise coming from the characteristic function in the de-nominator is negligible under our bandwidth conditions. Therefore, suchintermediate process Z˜n(r) that accounts for that estimation error will alsoapproximate the same tight Gaussian. Finally, our target process Z˜n(r) in-cludes estimation error coming from the variance term; therefore, if we showthat this estimation error is controlled then we will get the approximationto the same tight Gaussian.Step a) Gaussian approximation of the process Z∗n(r)Consider the class of functionsG ={(u, d) 7→ 1σn(r) h−1∫hh−1∫h∇λ21λ1λ2Re{e−iλ⊤(r−u)d−E [e−iλ⊤(r−∆Y)D]2π2ΨU0(λ)E[Di]}ΨK,H(λ)−2∑m=11πh−1∫h1λIm{e−iλ(rm−um)d−E [e−iλ(rm−∆Ym)D]Ψ∆Ym|D=0(λ)E[Di]}ΨK(λh)dλ : r ∈ I}By Lemma 5 and Lemma 6, we can obtain an envelope F of this classof functions given by|fr(·)| ≲ h2(αu−1)h−2αuh−1∫hh−1∫h1λ1λ2dλ1dλ2≲ h−2 log2 h−1.Let r1, r2 ∈ I be an ε-net of I. For any norm ∥·∥Q,2 from a discretedistribution Q, the brackets [fri − εF, fri + εF ] cover the class of functionsG, and they are of size 2ε∥F∥Q,2. This is a finite number of brackets due toI being a compact set. Thus we can apply Lemma 3 with A ∼ 1, v ∼ 1,σ = 1, b ∼ h−2 log2 h−1, q =∞ and γ = 1/ log n. From this result we claimthat there exists a random variable Wn with Wnd= supr∈I |ZGn (r)|, whereZGn (r) is a tight Gaussian in ℓ∞(I), such that∣∣∣∣supr∈I|Z∗n(r)| −Wn∣∣∣∣ = Op(log n log2/3 h−1n1/6h2/3)923.6. ProofsFurthermore, by Corollary 2.2.9 in Van der Vaart and Wellner (2023)E[supr∈I|ZGn (r)|]≲∫ 10√1 + log supQN (Gh, ∥·∥Q,2, ϵ)dϵ≲∫ 10√1 + log(A∥F∥P,2ϵ)dϵ≲√log h−1.The anticoncentration of the supremum of a Gaussian process yieldssupz∈R;∆>0P{z −∆ ≤ supr∈I |ZGn (r)| ≤ z +∆}∆≲ E[supr∈I|ZGn (r)|].Therefore,supz∈R|P{∥Z∗n∥I ≤ z} −P{∥ZGn ∥I ≤ z} ≲log n log2/3 h−1n1/6h2/3√log h−1 → 0,where the result comes from our assumption on that sequence.Step b) Estimation error from Ψˆ∆Y|D=0(λ)In this step we want to show that after estimating the characteristicfunction in the denominator, the intermediate processZ˜n(r) =√n(Fˆτ |D=1(r)−E[Fˆ ∗τ |D=1(r)])σn(r)also converges in distribution to the same tight Gaussian as in the previousstep. That is, we want to showsupz∈R|P{∥Z˜n∥I ≤ z} −P{∥ZGn ∥I ≤ z} → 0.We will do so by obtaining a result of the form ∥Z˜n−Z∗n∥I = op(an), forsome rate an → 0. By Lemma 6 it suffices to show that√nh2(αu−1) supr∈I|Fˆτ |D=1(r)− Fˆ ∗τ |D=1(r)| = op((log h−1)−1/2).The dominating term is the one with the double integral part; thus asufficient condition is thatsupr∈I∣∣∣∣∣∣∣12π2h−1∫hh−1∫h∇λ21λ1λ2Re{e−iλ⊤rΨˆU1(λ)}{ΨˆU0(λ)−1 −ΨU0(λ)−1}ΨK,H(λ)dλ1dλ2∣∣∣∣∣∣∣= op((log h−1)−1/2n−1/2h−2(αu−1)).933.6. ProofsTo tackle this problem, we divide the integral into four regionsRegion I: λ1 ∈ [h, λ0], λ2 ∈ [h, λ0]Region II: λ1 ∈ (λ0, h−1], λ2 ∈ [h, λ0]Region III: λ1 ∈ [h, λ0], λ2 ∈ (λ0, h−1]Region IV: λ1 ∈ (λ0, h−1], λ2 ∈ (λ0, h−1]We begin by bounding Region I. Write√n supr∈I12π2λ0∫hλ0∫h∇λ21λ1λ2Re{e−iλ⊤rΨˆU1 (λ)}{ΨˆU0 (λ)−1 −ΨU0 (λ)−1}ΨK(λ1)ΨK(λ2)dλ1dλ2≲√nλ0∫hλ0∫h∇λ21λ1λ2Re{e−iλ⊤r ΨˆU1 (λ)ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)ΨK,H(λ)dλ1dλ2≲√nλ0∫hλ0∫h∇λ2Re{ΨU1 (λ)ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)λ1λ2ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)ΨK,H(λ)dλ1dλ2+√nλ0∫hλ0∫h∇λ2Re{ΨˆU1 (λ)−ΨU1 (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)ΨK,H(λ)dλ1dλ2≲√nλ0∫hλ0∫h∇λ2Re{Ψτ (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)} ΨU0 (λ)ΨˆU0 (λ)ΨK,H(λ)dλ1dλ2+√nλ0∫hλ0∫h∇λ2Re{ΨˆU1 (λ)−ΨU1 (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)ΨK,H(λ)dλ1dλ2≲√n (1 + oP (1))λ0∫hλ0∫h∣∣∣∣ Ψτ (λ)ΨU0 (λ)∣∣∣∣ ·∣∣∣∣∣ ΨˆU0 (λ)−ΨU0 (λ)λ1λ2∣∣∣∣∣ dλ1dλ2+√n (1 + oP (1))λ0∫hλ0∫h∣∣∣ΨˆU1 (λ)−ΨU1 (λ)∣∣∣ ∣∣∣ΨˆU0 (λ)−ΨU0 (λ)∣∣∣ ∣∣∣∣ 1λ1λ2ΨU0 (λ)2∣∣∣∣ dλ1dλ2≲√n (1 + oP (1)){OP(log2 h−1n−1/2)+ λ0∫hλ0∫h(λ1λ2)−1∣∣∣ΨˆU1 (λ)−ΨU1 (λ)∣∣∣2 dλ1dλ21/2× λ0∫hλ0∫h(λ1λ2)−1∣∣∣ΨˆU0 (λ)−ΨU0 (λ)∣∣∣2 dλ1dλ21/2}≲√n (1 + oP (1)){OP(log2 h−1n−1/2)+OP (n−1) log2 h−1}≲ OP (log2 h−1 + log2 h−1n−1/2).943.6. ProofsThe results come from the fact that at low frequency the characteristicfunction in the denominator is bounded away from zero by a constant. Fur-thermore, the third to last inequality come from applying Cauchy-Schwarz.Next, notice that both Region II and Region III only differ in the orderof integration. Therefore, without loss of generality, we will bound RegionII by writing the following:√n supr∈I12π2λ0∫hh−1∫λ0∇λ21λ1λ2Re{e−iλ⊤rΨˆU1 (λ)}{ΨˆU0 (λ)−1 −ΨU0 (λ)−1}ΨK,H(λ)dλ1dλ2≲√nλ0∫hh−1∫λ0∇λ2Re{ΨˆU1 (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)ΨK,H(λ)dλ1dλ2≲√nλ0∫hh−1∫λ0∇λ2Re{ΨU1 (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)dλ1dλ2+√nλ0∫hh−1∫λ0∇λ2Re{ΨˆU1 (λ)−ΨU1 (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)dλ1dλ2≲√nλ0∫hh−1∫λ0∇λ2Re{Ψτ (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)} ΨU0 (λ)ΨˆU0 (λ)dλ1dλ2+√nλ0∫hh−1∫λ0∇λ2Re{ΨˆU1 (λ)−ΨU1 (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)dλ1dλ2≲√n (1 + oP (1))λ0∫hh−1∫λ0∣∣∣∣λ−11 λη2 Ψτ (λ)ΨU0 (λ)∣∣∣∣ ·∣∣∣∣∣ ΨˆU0 (λ)−ΨU0 (λ)λη+12∣∣∣∣∣ dλ1dλ2+√n (1 + oP (1))λ0∫hh−1∫λ0∣∣∣ΨˆU1 (λ)−ΨU1 (λ)∣∣∣ ∣∣∣ΨˆU0 (λ)−ΨU0 (λ)∣∣∣∣∣∣∣∣ 1λ1λ1−2αu2∣∣∣∣∣ dλ1dλ2≲√n (1 + oP (1)) λ0∫hh−1∫λ0λ2(η−αu)2λ1|Ψτ (λ)|2 dλ1dλ21/2× λ0∫hh−1∫λ0λ−2(η+1)2λ1∣∣∣ΨˆU0 (λ)−ΨU0 (λ)∣∣∣2 dλ1dλ21/2+ λ0∫hh−1∫λ0λ2αu−12λ1∣∣∣ΨˆU1 (λ)−ΨU1 (λ)∣∣∣2 dλ1dλ21/2953.6. Proofs× λ0∫hh−1∫λ0λ2αu−12λ1∣∣∣ΨˆU0 (λ)−ΨU0 (λ)∣∣∣2 dλ1dλ21/2≲√n (1 + oP (1)){O(hαu+γ−η−1/2 log1/2 h−1)OP (n−1/2hη+1/2 log1/2 h−1)+OP (n−1h−2αu log h−1)}≲ OP (hγ+αu+1/2 log1/2 h−1) +OP (n−1/2h−2αu log h−1).Finally, we bound Region IV as√n supr∈I12π2h−1∫λ0h−1∫λ0∇λ21λ1λ2Re{ΨˆU1 (λ)}{ΨˆU0 (λ)−1 −ΨU0 (λ)−1}ΨK,H(λ)dλ1dλ2≲√nh−1∫λ0h−1∫λ0∇λ2Re{ΨˆU1 (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)dλ1dλ2≲√nh−1∫λ0h−1∫λ0∇λ2Re{ΨU1 (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)dλ1dλ2+√nh−1∫λ0h−1∫λ0∇λ2Re{ΨˆU1 (λ)−ΨU1 (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)dλ1dλ2≲√nh−1∫λ0h−1∫λ0∇λ2Re{Ψτ (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)} ΨU0 (λ)ΨˆU0 (λ)dλ1dλ2+√nh−1∫λ0h−1∫λ0∇λ2Re{ΨˆU1 (λ)−ΨU1 (λ)λ1λ2ΨU0 (λ)}{ΨU0 (λ)− ΨˆU0 (λ)ΨU0 (λ)}ΨU0 (λ)ΨˆU0 (λ)dλ1dλ2≲√n (1 + oP (1))h−1∫λ0h−1∫λ0∣∣∣∣λη1λη2 Ψτ (λ)ΨU0 (λ)∣∣∣∣ ·∣∣∣∣∣ ΨˆU0 (λ)−ΨU0 (λ)λη+11 λη+12∣∣∣∣∣ dλ1dλ2+√n (1 + oP (1))h−1∫λ0h−1∫λ0∣∣∣ΨˆU1 (λ)−ΨU1 (λ)∣∣∣ ∣∣∣ΨˆU0 (λ)−ΨU0 (λ)∣∣∣ ∣∣∣∣ 1λ1λ2ΨU0 (λ)2∣∣∣∣ dλ1dλ2≲√n (1 + oP (1)) h−1∫λ0h−1∫λ0∣∣(λ1λ2)η−2αuΨτ (λ)∣∣2 dλ1dλ21/2× h−1∫λ0h−1∫λ0(λ1λ2)−2(η+1)∣∣∣ΨˆU0 (λ)−ΨU0 (λ)∣∣∣2 dλ1dλ21/2+ h−1∫λ0h−1∫λ0(λ1λ2)2αu−1∣∣∣ΨˆU1 (λ)−ΨU1 (λ)∣∣∣2 dλ1dλ21/2963.6. Proofs× h−1∫λ0h−1∫λ0(λ1λ2)2αu−1∣∣∣ΨˆU0 (λ)−ΨU0 (λ)∣∣∣2 dλ1dλ21/2≲√n (1 + oP (1)){O(h4αu+2γ−2η−1)OP (n−1/2h2η+1) +OP (n−1h−4αu )}≲ OP (h2γ+4αu ) +OP (n−1/2h−4αu ).Hence,√nh2(αu−1) supr∈I∣∣∣Fˆ ∗τ (r)− Fˆτ (r)∣∣∣ = OP (n−1/2h−2−2αu)= oP((log h−1)−1/2),which holds by Assumption BW.By this result, we have obtained that∥Z˜n − Z∗n∥I = op((log h−1)−1/2).This implies that there exists a sequence of constants ∆n ↓ 0 such thatP(∥Z˜n − Z∗n∥I > ∆n(log h−1)−1/2) ≤ ∆n,which follows from convergence in probability being metrized by the Ky Fanmetric (see Theorem 9.2.2 in Dudley (2002)). Now we can writeP(∥Z˜n∥I ≤ z)≤ P(∥Z∗n∥I ≤ z +∆n(log h−1)−1/2)+∆n≤ P(∥ZGn ∥I ≤ z +∆n(log h−1)−1/2)+ o(1)≤ P (∥ZGn ∥I ≤ z)+∆n (log h−1)−1/2 (log h−1)1/2= P(∥ZGn ∥I ≤ z)+ o(1),uniformly in z ∈ R, and where the second inequality comes from Step a,and the third inequality comes from the anticoncentration inequality. Like-wise, we also have P(∥Z˜n∥I ≤ z)≥ P (∥ZGn ∥I ≤ z)− o(1) uniformly. Thisconcludes this step.Step c) Estimation error from σˆn(r) and proof of the theorem.973.6. ProofsRecall thatKn(r) =12π21∫h21∫h2∇λ2Re(e−jλ⊤rλ1λ21Ψ∆Y|D=0(λ/h))ΨK(λ1)ΨK(λ2)dλ1dλ2.Once again, the dominating term is the one involving the double integralcomponent, and thus it is sufficient to show that the estimation error thereis controlled. In the upcoming discussions we derive the steps as if Kn wasthe only deconvolution kernel in the estimator.We then writesupu|Kn(u)| ≲ h−2αu∫ h−1h∫ h−1h1λ1λ2dλ1dλ2≲ h−2αu log2 h−1,Furthermore,∥Kˆn −Kn∥R2 ≤ supu∣∣∣∣∣ 12π2h−1∫hh−1∫h∇λ2Re(e−jλ⊤uλ1λ2(ΨˆU0(λ)−1 −ΨU0(λ)−1))×ΨK,H(λ)dλ1dλ2∣∣∣∣∣≲∣∣∣∣∣∣∣h−1∫hh−1∫h∣∣∣ΨˆU0(λ)−ΨU0(λ)∣∣∣λ1λ2ΨU0(λ)2ΨU0(λ)ΨˆU0(λ)dλ1dλ2∣∣∣∣∣∣∣≲ (1 + oP (1))h−1∫hh−1∫h(λ1λ2)2αu−1∣∣∣ΨˆU0(λ)−ΨU0(λ)∣∣∣ dλ1dλ2= OP(n−1/2h−4αu√log h−1).By combining these results we obtain∥Kˆ2n −K2n∥R2 ≤ ∥Kˆn −Kn∥R2∥Kˆn +Kn∥R2= Op(n−1/2h−4αu√log h−1)Op(h−2αu log2 h−1)= Op(h−6αun−1/2 log5/2 h−1),983.6. Proofswhich implies thatσˆ2n(r) =1nn∑i=1K2n(r−∆Yih)−[1nn∑i=1Kn(r−∆Yih)]2+Op(h−6αun n−1/2 log5/2 h−1).From the bound on Kn, Corollary A.1 in Chernozhukov et al. (2014) andTheorem 2.14.1 in Van der Vaart and Wellner (2023) we can verify that1nn∑i=1K2n(r−∆Yih)= E[K2n(r−∆Yih)]+Op((h−2αu log2 h−1)2n−1/2),1nn∑i=1Kn(r−∆Yih)= E[Kn(r−∆Yih)]+Op(h−αu log2 h−1n−1/2),uniformly in r ∈ I. Therefore, we writeσˆ2n(r) = σ2n(r) +Op(n−1/2[h−4αu log4 h−1 + h−6αu log5/2 h−1]),so thatσˆ2n(r)σ2n(r)= 1 +O(h2(2αu−1))Op(n−1/2[h−4αu log4 h−1 + h−6αu log5/2 h−1])= 1 +Op(n−1/2h−2 log4 h−1 + n−1/2h−2−2αu log5/2 h−1)= 1 + op((log h−1)−1),provided by Assumption BW.By the results in the previous steps (in particular the bound ∥ZGn ∥I =Op(√log h−1n)), we can show that ∥Z˜n∥I = Op(√log h−1n). Therefore, wewrite∥Zˆn − Z˜n∥I ≤ ∥σn/σˆn − 1∥I∥Z˜n∥I = op((log h−1n)−1/2).By the same arguments as the previous step we can conclude thatsupz∈R∣∣∣P(∥Z˜n∥I ≤ z)−P (∥ZGn ∥I ≤ z)∣∣∣→ 0.Thus the proof is complete. ■993.6. Proofs3.6.4 Proof of Theorem 3Step a)Recall the definition of the multiplier process in Equation (3.17). Wedefine an infeasible version of this process, given byZξn(r) =1σn(r)√nn∑i=1ξi{L1,n(r1 −∆Y1,ih)− 1nn∑l=1L1,n(r1 −∆Y1,lh)+ L2,n(r2 −∆Y2,ih)− 1nn∑l=1L2,n(r2 −∆Y2,lh)(3.17)+Kˆn(r1 −∆Y1,ih,r −∆Y2,ih)− 1nn∑l=1Kˆn(r1 −∆Y1,lh,r −∆Y2,lh)},The goal of this step is to show that this process approximates the sametight Gaussian in ℓ∞(I) as in Theorem 2. To do so, recall the class Gndefined in Step a) of the proof of Theorem 2, and letνn(g) =1√nn∑i=1ξ{g(∆Yi)− 1nn∑l=1g(∆Yl)}, g ∈ Gn.From Lemma 3 applied to ||νn||Gn with b ∼ h−2 log2 h−1, σ = 1, A ∼ 1,v ∼ 1, γ = 1/ log n and q sufficiently large yields that there exists a randomvariable W ξn of which the conditional distribution given Dn is the samedistribution of ||Gn||Gn (= ||ZGn ||I), that is,P{W ξn ≤ z | Dn} = P{||ZGf,n||I ≤ z},for all z ∈ R almost surely, and such that∣∣||νξn||Gn −W ξn∣∣ = op{(log h−1n )−1/2}.This means that there exists a sequence of constants ∆n ↓ 0 such thatP{||νξn||Gn −W ξn > ∆n (log h−1n )−1/2 | Dn} p→ 0by Markov’s inequality. Since ||νξn||Gn = ||Zξf,n||I we have thatP{||Zξn||I ≤ z | Dn} ≤ P{W ξn ≤ z +∆n (log h−1)−1/2 | Dn}+ op(1)= P{||ZGn ||I ≤ z +∆n (log h−1)−1/2}+ op(1)= P{||ZGn ||I ≤ z}+ o(1),1003.6. Proofsuniformly in z ∈ R, and where the last line comes from the anti-concentrationproperty of supremum of Gaussian processes.Similarly, we haveP{||Zξn||I ≤ z | Dn} ≥ P{||ZGn ||I ≤ z}− o(1),uniformly in z ∈ R. Therefore, we obtain the conclusion of this step.Step b)Based on the previous step it suffices to show that||Zˆξn − Zξn||I = op{(log h−1)−1/2}.To do so, we first define an intermediate process given by the followingexpression:Z˜ξn(r) =1σn(r)√nn∑i=1ξi{Lˆ1,n(r1 −∆Y1,ih)− 1nn∑l=1Lˆ1,n(r1 −∆Y1,lh)+ Lˆ2,n(r2 −∆Y2,ih)− 1nn∑l=1L2,n(r2 −∆Y2,lh)(3.18)+Kˆn(r1 −∆Y1,ih,r −∆Y2,ih)− 1nn∑l=1Kˆn(r1 −∆Y1,lh,r −∆Y2,lh)},To proceed, notice that||Zˆξn − Zξn||I ≤ ||Zˆξn − Z˜ξn||I + ||Z˜ξn − Zξn||I .We’ll start with the bound of the second term. From step b) in the proofof Theorem 2 we have that the dominating term satisfies∣∣∣∣∣∣∣∣ 1σn√nn∑i=1{Kˆn( · −∆Yih)− 1nn∑l=1Kˆn( · −∆Ylh)}∣∣∣∣∣∣∣∣I= op((log h−1)−1/2).Now, since ∥1/σn∥I ≲ h2αu−2, it suffices to prove that∣∣∣∣∣∣∣∣ n∑i=1ξi{Kˆn( · −∆Yih)− 1nn∑j=1Kˆn( · −∆Yjh)}∣∣∣∣∣∣∣∣I= op{h−2α+2√n(log h−1)−1/2}.1013.6. ProofsObserve that∣∣∣∣ n∑i=1ξi{Kˆn( · −∆Yih)− 1nn∑l=1Kˆn( · −∆Ylhn)}∣∣∣∣≲∫R∫R∣∣∣∣∣n∑i=1ξieiλ⊤∆Yi/hλ1λ2∣∣∣∣∣∣∣∣∣ 1Ψˆ∆Y|D=0(λ/h) − 1Ψ∆Y|D=0(λ/h)∣∣∣∣|ΨK,H(λ)|dλ1dλ2≲{∫R∫R1λ1λ2∣∣ n∑i=1ξieiλ⊤λi/h∣∣2|ΨK,H(λ)|dλ1dλ2}1/2×{∫R∫R1λ1λ2∣∣∣∣ 1ΨˆU0(λ/hn) − 1ΨU0(λ/h)∣∣∣∣2|ΨK,H(λ)|dλ1dλ2}1/2= Op{n1/2n−1/2h−2αu log2 h−1},Therefore,∥Z˜ξn − Zξn∥I ≲ OP(n−1/2h−2(1−2αu)h−2αuh−2)≲ OP(n−1/2h−4+2αu)= op{(log h−1n)−1/2},given by Assumption BW.Now, recall that from step a) in the proof of Theorem 2 we obtained that∥σn/σˆn−1∥I = Op{(log h−1)−1}, and observe that ||Z˜ξn|| = Op(√log h−1)because E[||ZGn ||I] = O (√log h−1). Then||Zˆξn − Z˜ξn||I = op{(log h−1)−1/2}.Combining both results yields that||Zˆξf,n − Zξf,n||I = op{(log h−1n)−1/2},which leads tosupz∈R∣∣P{||Zˆξf,n||I ≤ z | Dn}− P{||ZGf,n||I ≤ z}∣∣→ 0,by the same arguments as in the proof of Theorem 2.Step c)In this step, we shall verify that the quantiles using the multiplier processhave asymptotic validity and we will find the rate of supremum width of theconfidence band Cˆn.1023.6. ProofsThe end result of the previous step implies that there exists ∆n ↓ 0 suchthat, with probability greater than 1−∆n,supz∈R∣∣P{||Zξn||I ≤ z | Dn} − P{||ZGn ||I ≤ z}∣∣ ≤ ∆n. (3.19)Taking ∆n ↓ 0 more slowly if necessary, we also have thatsupz∈R∣∣P{||Zˆn||I ≤ z | Dn} − P{||ZGn ||I ≤ z}∣∣ ≤ ∆n.Let En denote the event where 3.19 holds, and let cGn (u) denote the u-quantile of ||ZGn ||I , for u ∈ (0, 1). Then on the event En we have thatP{||Zˆξn||I ≤ cGn (1− α+∆n) | Dn} ≥ P{||ZGn ||I ≤ cGn (1− α+∆n)}−∆n= 1− α,because ||ZGn ||I has a continuous distribution. Then we have the inequalitycˆn(1− α) ≤ cˆGn (1− α+∆n),which holds on En.Next, we writeP{||Zˆn|| ≤ cˆn(1− α)} ≤ P{||Zˆn|| ≤ cGn (1− α+∆n)}+ P (Ecn)≤ P{||ZGn || ≤ cGn (1− α+∆n)}+ 2∆n= 1− α+ 3∆n.Similarly,P{||Zˆn|| ≤ cˆn(1− α)} ≥ P{||Zˆn|| ≤ cGn (1− α−∆n)}− P (Ecn)≤ P{||ZGn || ≤ cGn (1− α−∆n)}− 2∆n= 1− α− 3∆n.Thus we obtain thatP{||Zˆn|| ≤ cˆn(1− α)}→ 1− α,as desired.Finally, to obtain the width of Cˆn we use the Borell-Sudakov-Tsirelsoninequality (Van der Vaart and Wellner (2023) Lemma A.2.2), which leadstocGn (1− α) ≤ E[||ZGn ||I]+√2 log(1/(α−∆n) ≲√log h−1,1033.6. Proofswhich implies that cˆn(1− α) = Op(√log h−1). Furthermore,supr∈Iσˆn(r) ≤ supr∈Iσn(r) · supr∈Iσˆn(r)σn(r)= Op{h−2α+2}.Therefore, the supremum width of the band Cˆn is2 supr∈Iσˆn(r)√n≤ 2√nsupr∈Iσn(r) · supr∈Iσˆn(r)σn(r)Op(√log h−1)= Op(√log h−1h−2α+2√n).This completes the proof. ■1043.6. Proofs3.6.5 Proof of Corollary 1By leveraging trigonometric properties of sines and cosines of sums and usingarguments similar to Hall and Lahiri (2008), we obtain the result∥Fτ −E[F ∗τ ]∥I = O(h2γ).Next, the assumption implies√n (Fτ (r)− Fτ (r))σˆn(r)=[1 + oP {(log h−1)−1}] √n (Fτ (r)− Fτ (r))σn(r)= Z˜n(r) + o{(log h−1)−1/2},uniformly over r ∈ I. Furthermore, Z˜n is the same intermediate processdescribed in Step b) of the proof of Theorem 2. Using results from the proofof the Theorem just mentioned, in particular the result on the Gaussian ap-proximation of ∥Z˜n∥I derived in Step b), we can apply the anti-concentrationinequality.This leads to the following expression:supz∈R2|P {∥√n (Fτ (r)− Fτ (r)) /σˆn(r)∥I ≤ z}− P {∥ZGn ∥I ≤ z}| → 0.Finally, due to the same arguments as Step c) in the proof of Theorem 2,we obtain the desired result. ■105BibliographyJohn M Abowd, Francis Kramarz, and David N Margolis. High wage workers andhigh wage firms. 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From characteristic function to distribution function. EconometricTheory, 1991.Gabriel Ulyssea. Firms, informality, and development: Theory and evidence frombrazil. American Economic Review, 108(1):2015–2047, 2018.Gabriel Ulyssea. Informality: Causes and consequences for development. AnnualReview of Economics, 12, 2020.Aad Van der Vaart and Jon A. Wellner. Weak convergence and empirical processes.Springer Series in Statistics, 2023.109Appendix AAppendix to Chapter 1A.1 Additional FiguresFigure A.1: Macroeconomic Variables between 2011 and 2017(a) Gross Domestic Product (b) Unemployment(c) InflationNote: This figure shows some macroeconomic indicators around the time period of analysis.Panel (a) shows the nominal GDP, Panel (b) the unemployment rate, and Panel (c) shows theinflation rate. Source: World Bank.110A.1. Additional FiguresFigure A.2: Employment Effects, Extensive and Intensive Margins(a) Intensive and Extensive Margin(b) Excluding Extensive MarginNote: This figure shows the estimates from Equation (1.5) using firm employment as the outcomevariable. Panel (a) includes firms that closed after the minimum wage increase and thus recordzero employment, and Panel (b) excludes these firms. Regressions are weighted by the logarithmof total employment in 2016, and robust standard errors are computed. Point estimates and their95% confidence intervals are shown in this figure. The vertical red line corresponds to the baselineyear, prior to the minimum wage hike. Source: PLAME.111A.1. Additional FiguresFigure A.3: Employment Effects, by Sector: Firm Approach(a) Manufacture (b) Commerce(c) Restaurants (d) ConstructionNote: This figure shows the estimates from Equation (1.5) using firm employment as the outcomevariable. The different panels compute the estimates within the subset of firms in a particularindustry. Regressions are weighted by the logarithm of total employment in 2016, and robuststandard errors are computed. Point estimates and their 95% confidence intervals are shown inthis figure. The vertical red line corresponds to the baseline year, prior to the minimum wagehike. Source: PLAME.112A.1. Additional FiguresFigure A.4: Composition of Labor Force in PeruNote: This figure shows the employment status composition of the labor force in Peru aroundthe years of analysis. The orange area corresponds to workers in the informal sector, more than80% of that area corresponds to self-employment. Source: ENAHO.113A.1. Additional FiguresFigure A.5: Number of Firms in EEA across provincesNote: This figure shows the geographical (province-level) distribution of firms in the EEA dataset.Darker colours indicate larger number of firms in that province. Source: EEA.114A.1. Additional FiguresFigure A.6: Linearity between Exposure and Wage GrowthNote: This figure shows the slope between the exposure variable in Equation (1.5) and changesin average wages (dependent variable). A non-linear relationship would threat the identificationstrategy of this approach.Figure A.7: Linearity between Exposure and Employment GrowthNote: This figure shows the slope between the exposure variable in Equation (1.5) and em-ployment changes (dependent variable). A non-linear relationship would threat the identificationstrategy of this approach. Source: PLAME.115A.1. Additional FiguresFigure A.8: Own-Wage Elasticity in the LiteratureNote: This figure shows the implied own-wage elasticities of my reduced form estimates basedon Equation (1.5). To construct confidence intervals I jointly estimate the regression using bothoutcomes (i.e. stacking influence functions). The graph compares these elasticities to the list ofpapers shown in Harasztosi and Lindner (2019). Source: PLAME.116A.1. Additional FiguresFigure A.9: Out-of-sample prediction of Fraction Affected (FA)Note: This figure shows the slope of the regression between the actual fraction affected andthe predicted one, based on a regression forest model used on a training sample. This plot isconstructed using a sample not used for training the regression model. Source: PLAME.117A.1. Additional FiguresFigure A.10: Effects on Type of Contract: Worker ApproachNote: This figure plots the effects on job stability using the worker approach described in Section1.7. The dependent variable is an indicator equal to 1 if the worker has a permanent contract.The upper panel plots the coefficients from Equation (1.7) as of February 2016 (pre-policy period)and February 2017 (post-policy period). The lower panel shows the difference between the blueand red line in the upper panel, which correspond to the coefficients of Equation (1.8). Source:PLAME.118A.1. Additional FiguresFigure A.11: Effects on J2J Transition: Worker ApproachNote: This figure plots the effects on job-to-job transitions using the worker approach describedin Section 1.7. The dependent variable is an indicator equal to 1 if the worker separated from hisold firm and remained formally employed. The upper panel plots the coefficients from Equation(1.7) as of February 2016 (pre-policy period) and February 2017 (post-policy period). The lowerpanel shows the difference between the blue and red line in the upper panel, which correspond tothe coefficients of Equation (1.8). Source: PLAME.119A.1. Additional FiguresFigure A.12: Effects on Firm Size: Worker ApproachNote: This figure plots the effects on firm quality using the worker approach described in Section1.7. The dependent variable is the change in firms’ size. The upper panel plots the coefficients fromEquation (1.7) as of February 2016 (pre-policy period) and February 2017 (post-policy period).The lower panel shows the difference between the blue and red line in the upper panel, whichcorrespond to the coefficients of Equation (1.8). Source: PLAME.120A.2. Data AppendixA.2 Data AppendixIn this appendix I discuss some details that were left out in the main dis-cussion of the paper.A.2.1 Linking PLAME and EEAAs mentioned in the main text, there is no way to link both datasets basedon the firm identifier. Instead, I opt to train a predictive model in thePLAME dataset to predict out-of-sample the fraction affected in the EEAdataset.Both datasets share a set of observables for each year throughout 2015to 2018:• Total employment among males and females in each of these categories:managers, permanent employees, temporary employees, permanent la-borers (obreros), and temporary laborers.• Total wage cost among permanent and temporary workers• Industry group• Date of firm creationThe goal is to fit a prediction model, using these variables, in the PLAMEdataset and then make an out-of-sample prediction on the EEA dataset. Asimilar exercise is performed in Harasztosi and Lindner (2019), where theyshow that a quadratic polynomial in average wage cost per worker is highlypredictive of the fraction affected variable.In this paper, I choose to leverage existing machine learning methodsto perform this step. First, I draw a random 75% sample of the PLAMEdataset to be used as training data to predict FAj on the firm-level census.The results of the prediction model are shown in Figure A.9. In particular,it plots a linear fit between the actual FA of the remaining 25% PLAMEsample, and the predicted estimates. The slope between both is very closeto one, which suggests that the prediction model has performed sufficientlywell for my purposes.A.2.2 Firm concentration observed in EEAFigure A.5 shows the firm counts for each province of the EEA dataset.As can be observed in the map, there is a high degree of concentration in121A.2. Data Appendixmany locations, mainly outside of the coastal (left) area. If we excluded thepresence of the informal sector in the analysis we would - perhaps incorrectly- assume that these firms exert a lot of market power. However, if these firmsare competing with the informal sector, it could be the case that the de-factomarket power is smaller than expected.122A.3. Additional Empirical EvidenceA.3 Additional Empirical EvidenceIn this section I show additional empirical evidence not discussed in themain text.Job Stability In Figure A.10 I show the results of running the worker-approach on a dependent variable equal to 1 if the worker holds a perma-nent contract. The results show that, conditional on remaining formallyemployed, workers were less likely to hold a permanent contract after theminimum wage increase. This suggests that this policy can perpetuate acycle of being trapped within jobs with no stability.Combining these results with the previous evidence in the main text,it would suggest that some workers separate from their firms and land insimilar-quality firms but on a temporary contract instead. Temporary con-tracts are a common feature in the context of Peru since a reform enactedin 2002, and this might be a potentially important non-pecuniary channelthat can negatively impact welfare.Reallocation In Figure A.11 I provide additional evidence that the real-location channel (Dustmann et al., 2022) is muted in my context. This figureplots the estimated coefficients on the probability of transitioning from jobto job within formal employment. In particular, by comparing with the em-ployment effects in Figure 1.5, it suggests that workers are exiting formalemployment rather than reallocating across firms. Additionally, Figure A.12plots the effects on firm size (conditional on formal employment), and showsthat low-wage workers are not switching to larger firms.123A.4. Upper Tail ImputationA.4 Upper Tail ImputationThe results are robust to imputing the upper tail of the earnings distribution.To address censoring at the 95th percentile of each year, I can impute uppertail earnings following Card et al. (2013).In particular, I create 10-year age cells (20 to 29, 30 to 39, . . . , 50 to59), and 6 education cells (missing, no qualifications, secondary, some postsec, univ graduate, post graduate). I then construct the mean log-earningsof individual i in all other periods, and for all their coworkers. For singletonworkers or singleton firms I use the sample mean of gender g(i). Finally, Ifit a series of Tobit models separately by year, gender, education level, andage range cells that include the following variables: age, mean log earnings,in other years, fraction of censored earnings in other years, number of full-time employees of gender g and its square, dummy for 11 or more employees,fraction of univ graduates at the firm, mean log wage co-workers and fractionof coworkers with censored earnings, dummy for singleton individuals, anda dummy for employees of 1-worker firms.If y ∼ N(X ′β, σ) and censoring is such that y ≥ c is censored. Let k =Φ [(c−X ′β)/σ)], where Φ(·) is the standard normal CDF. Let u ∼ U [0, 1],thenyu = X ′β + σΦ−1 [k + u(1− k)] .With the estimates βˆ and σˆ from the Tobit models along with randomdraws from the uniform distribution supported on the unit interval, I canthen impute yˆu on the censored observations in my dataset.124A.5. AKM EffectsA.5 AKM EffectsIn this section I provide additional details to the construction of the AKMeffects used in Table 1.7 to characterize the impacts of the minimum wageon reallocation. Using the PLAME data for the years 2015 and 2016 (priorto the minimum wage increase), I estimate a model of log earnings againsta cubic polynomial in age and firm fixed effects and then residualize theoutcome.Then, I estimate an AKM model of the form:yit = αi +ΨJ(i,t) + λt + uit, (A.1)where yit are log-earnings; αi represent a time-invariant ability componentof workers; J(i, t) is a function that returns the firm of worker i at time t,so that ΨJ(i,t) represent firm-specific premia; λt are time fixed effects.Identification in the AKM model relies on the existence of worker tran-sitions across firms, and requires these movements to be uncorrelated withtime-varying residual components of earnings (conditional on worker andfirm fixed effects).The estimated firm fixed effects are labelled AKM effects and measurean employer-specific pay premium shared across all workers at that firm.That estimates of the firm and worker effects are identified within aconnected set of firms that are linked through by workers’ moves. The initialsample has 5, 515, 960 worker-year observations and the largest connectedset restricts the sample to 4, 429, 096 observations.125Appendix BAppendix to Chapter 2B.1 Additional FiguresFigure B.1: Effect on Household’s Consumption Price IndexNote: This figure plots the effect of the minimum wage on a Laspeyres price index of foodconsumption at the household level. These regressions are run on a household-level panel fromthe ENAHO dataset. Point estimates and their 95% confidence intervals are shown. Source:ENAHO.126B.1. Additional FiguresFigure B.2: Expected gains from MWNote: This figure plots the expected gains from the minimum wage as described in Section B.2Point estimates and their 95% confidence intervals are shown. Source: PLAME and ENAHO.127B.2. Alternative Computation of the Gains of the Minimum WageB.2 Alternative Computation of the Gains of theMinimum WageIn this section I explore an alternative approach to compute the gains of theminimum wage shown in Figure 2.2 of the main text.First, using the ENAHO dataset, I estimate the relationship with formalearnings at baseline and the logarithm of income in the informal sector inthe next year. I use the subset of individuals who transition from formalemployment to informal self-employment and estimate the following model:log(informal incomei,t) =8∑b=1γ2016,b1{formal earningsi,t−1 ∈ binb}+ λsector(i,t-1) + ϵi,t, (B.1)where λsector(i,t-1) are industry group fixed effects at baseline year t− 1.I then use the estimated coefficients to predict, in the PLAME dataset,the counterfactual income in the informal sector among those who werecharacterized by transitioning from employment to non-employment in thisdataset.Finally, I estimate the income effects by running Equation (1.8) on theimputed logarithm of income produced in the previous step. Figure B.2plots the estimates across different wage bins. While the coefficient in the[850, 950) bin is small and negative, these results do not qualitatively changethe main results.128
Thesis/Dissertation
eng
Economics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Saggio, Raffaele
Essays on labor economics and applied econometrics
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