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Three essays in applied economics Dai, Guang 2013

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Three Essays in Applied EconomicsbyGuang DaiB.E., Harbin Engineering University, 2001M.A., Nanjing University, 2004M.A., Universite? de Toulouse, 2007A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate Studies(Economics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2013c? Guang Dai 2013AbstractThis dissertation discusses three topics in applied economics.The first essay examines the causal effect of social capital on individ-ual income by exploiting the historically determined pattern of family namedistribution in Chinese villages. Family name distribution impacts socialcapital through historical inter-lineage rivalry and cooperation. The esti-mates show a strong first order effect on male villagers, which implies aone standard deviation increase in social capital is equivalent to two to fouryears of education. No effects on female villagers were found. The genderdifferentiation could be accounted for by occupation difference: male vil-lagers? income mainly comes from market exchange, while female villagers?income comes mainly from home production. Using a simple model, it isdemonstrated that a village?s social capital determines its trade scope andtherefore income of its residents.The second essay proposes a general method to identify subjective ex-pectation bias. The method exploits an implication of rational expectationsthat requires the identical weight of an independent variable in projectingboth objective and subjective probabilities. The empirical analysis showsthat female seniors do not correctly internalize age information while maleseniors fail at internalizing income information. Though cognitive abilityand risk aversion can partially explain the results, they are not the sourcesof the identified biases.The third essay explores how seniors make long term care insurance(LTCI) decisions by developing a dynamic structural discrete choice modelwhere a rational, risk averse, bequest motivated senior has to decide at eachperiod whether to buy an insurance policy or not. Using the Health and Re-tirement Survey data, this essay finds substantial heterogeneity in bequestiiAbstractmotive that drives LTCI decisions. Specially, the idiosyncratic bequest mo-tive helps to explain why LTCI holders do not experience a higher incidencerate than non-holders.iiiPrefaceThis dissertation is original, unpublished, independent work by the author,Guang Dai.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Family Name Distribution: The Effect of Social Capital onIncome . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Motivating Theory and Estimation Framework . . . . . . . . 82.2.1 Motivating Theory . . . . . . . . . . . . . . . . . . . 82.2.2 Estimation Framework . . . . . . . . . . . . . . . . . 112.3 Data and OLS Estimates . . . . . . . . . . . . . . . . . . . . 142.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 OLS Estimates . . . . . . . . . . . . . . . . . . . . . . 192.4 First Stage Analysis . . . . . . . . . . . . . . . . . . . . . . . 212.4.1 Village, Lineage and Family Name . . . . . . . . . . . 222.4.2 Inter-Lineage Rivalry and Cooperation . . . . . . . . 25vTable of Contents2.4.3 Measurement . . . . . . . . . . . . . . . . . . . . . . . 282.4.4 First Stage Analysis . . . . . . . . . . . . . . . . . . . 312.4.5 First Stage Robustness . . . . . . . . . . . . . . . . . 332.5 IV Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.5.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . 392.5.2 Robustness to Weak IV . . . . . . . . . . . . . . . . . 422.5.3 Robustness to Too Much Noise . . . . . . . . . . . . . 442.5.4 The Exclusion Assumption . . . . . . . . . . . . . . . 452.5.5 Interpretation of the Gender Difference . . . . . . . . 472.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 Identifying Subjective Expectation Bias: Method and Evi-dence from the Health and Retirement Study . . . . . . . . 513.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 513.2 Definition and Identification Assumption . . . . . . . . . . . 543.2.1 A Belief Literature Review . . . . . . . . . . . . . . . 543.2.2 Definition . . . . . . . . . . . . . . . . . . . . . . . . . 573.2.3 Identification Assumption . . . . . . . . . . . . . . . . 613.2.4 Model and Test Method . . . . . . . . . . . . . . . . 643.3 Background, Data and Description Analysis . . . . . . . . . 663.3.1 HRS and Expectation Question . . . . . . . . . . . . 663.3.2 Choosing Covariates . . . . . . . . . . . . . . . . . . . 683.3.3 Descriptive Statistics . . . . . . . . . . . . . . . . . . 713.3.4 Assessing Subjective Probability . . . . . . . . . . . . 743.4 Age Bias and Income Bias . . . . . . . . . . . . . . . . . . . 783.4.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . 783.4.2 Threats to the Main Results . . . . . . . . . . . . . . 853.5 Sources of Biases . . . . . . . . . . . . . . . . . . . . . . . . . 903.5.1 Cognitive Ability . . . . . . . . . . . . . . . . . . . . 913.5.2 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . 933.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94viTable of Contents4 Risk Aversion vs. Bequest Motive: How do Seniors MakeLong Term Care Insurance Decisions? . . . . . . . . . . . . 974.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 Institution Background . . . . . . . . . . . . . . . . . . . . . 1004.2.1 The Eligibility Rules and Estate Recovery Policy ofMedicaid . . . . . . . . . . . . . . . . . . . . . . . . . 1004.2.2 Long Term Care Insurance Market . . . . . . . . . . 1024.3 Data and Primary Analysis . . . . . . . . . . . . . . . . . . 1034.4 Model: Specification, Identification and Estimation . . . . . 1074.4.1 Specification . . . . . . . . . . . . . . . . . . . . . . . 1084.4.2 Preference Identification . . . . . . . . . . . . . . . . 1174.4.3 Belief Estimate and Parameters Calibration . . . . . 1224.4.4 Likelihood Function . . . . . . . . . . . . . . . . . . 1294.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.5.1 The Distribution of ? and ? . . . . . . . . . . . . . . 1314.5.2 Evaluating ex post Preferences . . . . . . . . . . . . . 1324.5.3 Understanding the Puzzle . . . . . . . . . . . . . . . . 1374.5.4 Counterfactual Policy Analysis . . . . . . . . . . . . . 1394.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147viiList of Tables2.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . 202.2 Effect of Social Capital on Income: OLS Estimates . . . . . . 212.3 Families Name Composition and Social Capital . . . . . . . . 292.4 First Stage Regression for Social Capital . . . . . . . . . . . . 342.5 First Stage Estimates: Minority Villages vs. Han Villages . . 362.6 First Stage Estimates: South vs. Non-south . . . . . . . . . . 372.7 Effect of Social Capital on Income: IV Estimates . . . . . . . 412.8 Effect of Social Capital on Income: Robust to Weak IV . . . 432.9 Effect of Social Capital on Income: Randomization Test . . . 452.10 Does Family Name Distribution Affects Public Goods Provi-sion? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.1 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . 723.2 Can Subjective Probability Predict Behaviour? . . . . . . . . 773.3 Main Results: Male Sample . . . . . . . . . . . . . . . . . . . 793.4 Main Results: Female Sample . . . . . . . . . . . . . . . . . . 803.5 Assessing the Measurement Error due to Cognitive Ability . . 883.6 Is Age Bias a Result of Wishful Thinking? . . . . . . . . . . . 893.7 Are Cognitive Factors the Source of Biases? . . . . . . . . . . 923.8 Is Risk Aversion the Source of Biases? . . . . . . . . . . . . . 954.1 Descriptive Statistics by LTC Insurance Purchase Status . . . 1054.2 Bequest Probability and Risk Aversion by Insurance Coverageand Risk Incidence Status . . . . . . . . . . . . . . . . . . . . 1074.3 Preference Distribution Estimates . . . . . . . . . . . . . . . . 1324.4 How Bequest Motive Predicts Subjective Bequest Probability? 137viiiList of Tables4.5 Positive Correlation Test . . . . . . . . . . . . . . . . . . . . . 1394.6 Policy Effects under Various Scenarios . . . . . . . . . . . . . 1414.7 LTCI Purchasing Probability across Different Groups Beforeand After Policy Changes . . . . . . . . . . . . . . . . . . . . 143ixList of Figures2.1 Trust Level in Different Groups . . . . . . . . . . . . . . . . . 162.2 Fractionalization vs. Polarization Index . . . . . . . . . . . . 322.3 Partial Correlation of Social Capital and Instruments . . . . . 323.1 Comparing Objective and Subjective Probabilities: by Nurs-ing Home Experience . . . . . . . . . . . . . . . . . . . . . . . 743.2 Comparing Objective and Subjective Probabilities: by LongTerm Care Insurance Holding . . . . . . . . . . . . . . . . . . 753.3 Comparing Objective and Subjective Probabilities: by Age . 754.1 Optimal Decision Without Utility Shocks . . . . . . . . . . . 1194.2 Optimal Decision With Utility Shocks . . . . . . . . . . . . . 119xAcknowledgementsI would like to express my sincere gratitude to my advisor Prof. KevinMilligan for the continuous support of my Ph.D study and research, for hispatience, motivation, encouragement, and kindness. His guidance helpedme in all the time of research and writing of this dissertation.Besides my advisor, I would like to thank the rest of my thesis committee:Prof. Hiro Kasahara and Prof. Marit Rehavi. It is almost impossiblewithout the Hiro?s help. Thanks also go to Prof. Siwan Anderson, Prof.Thomas Lemieux, Prof. Thomas Davidoff and Prof. Steven Lehrer for theirinsightful comments and hard questions.It was a great pleasure to spend four years with my classmates: DaveFreeman, Xiaodan Gao, Mingzhi Wang, Jinwen Xu, Donna Feir, MustafaTugan, Terry Keenland and Andrew Hill. Dave and Xiaodan helped a lotwhile I was away from the UBC in the past two years.My colleagues at MSA gave me full support to finish the dissertation.Special thanks go to Harry Chandler, Richard Penn, Mike Nozdryn-Plotnicki,Matt Ayres, Doug Doll and Donna Ehrhardt.Last, I want to thank my families for their support and love. Thank you,Lucy, for all the time spent together, good or bad.xiDedicationTo my wife, Lucy Danxia Mao, and daodaosxiiChapter 1IntroductionThis dissertation consists of three essays in applied economics.The first essay examines the causal effect of social capital on individualincome by exploiting the historically determined pattern of family name dis-tribution in Chinese villages. Though social capital has been reported to as-sociate with improved government quality (Putnam, Leonardi, and Nanetti[1993]), better financial development (Guiso, Sapienza and Zingales [2004])and increased economic growth (Knack and Keefer [1997]) etc., to build acausal argument for social capital upon individual economic achievementfaces several challenges, which include measurement and endogeneity prob-lems. This essay employs the rural part of the Chinese General Social Surveyto address these problems and exploits a unique feature of the Chinese cul-ture to build causality.Unlike villages in other societies, a Chinese rural village generally hasonly a few different surnames among its residents. Families with the samesurname are usually from the same ancestor who first settled down at thelocality hundreds of years ago. Because of either practical needs or theConfucian philosophy, these families with the same surname were united asa lineage or family clan. In the long run of histo, the interaction betweenthese lineages, which can be rivalry or cooperation depending on resourceconstraints and objective functions, has a lasting impact on social capital.By instrumenting social capital with historical rivalry or cooperation amonglineages, this essay is capable of pinning down the desired causal effect.The second essay proposes a general method to identify subjective expec-tation bias. It exploits an implication of rational expectations that requiresthe identical weight of an independent variable in projecting both objec-tive and subjective probabilities (Manski [2004]). To illustrate, consider the1Chapter 1. Introductionproblem of seniors in the Health and Retirement Study (HRS) data set.The seniors were asked to assess their individual probability of entering anursing home within five years. Consider the information represented bya variable reporting one?s health status. Under the rational expectationsassumption, the coefficient of health status should be ? in projecting thesubjective probability, if the coefficient in projecting objective probabilitywere ?.Of course, the objective probability could never be known outside a con-trolled experimental environment. What is observed is the realization of arandom event. Thus, to assess rationality and identify expectation bias, it isnecessary to substitute the observed realizations for objective probabilities.The substitution brings an identification issue, which is whether the coef-ficients are still comparable between the equations of objective probabilityand of ex post realization, by introducing interim events between the forma-tion of expectations and the realization of random events. This essay showsthat as long as an orthogonality condition holds, the identification can besuccessfully achieved.The last essay explores how seniors shop for long term care insurance(LTCI) policies. It is of importance because of two issues of the market.First, the LTCI market is very thin. Though a senior?s lifetime chanceof using long term care is more than 40% (Kemper et al. [1991]), onlyabout 10% of seniors are covered by any LTCI policies. Second, Accordingto classical asymmetric information theory, the LTCI market is expectedto observe a positive correlation between insurance coverage and incidentoccurrence due to adverse selection and moral hazard. However, recentresearch noticed that in the LTCI market those covered by an insurancepolicy do not experience higher incidence rate than those not (Finkelsteinand MaGarry [2006]).To explore underlying preferences that drive the LTCI shopping deci-sions, this essay develops a dynamic structural discrete choice model wherea rational, risk averse, bequest motivated senior has to decide at each periodwhether to buy an insurance policy or not. In this model, both risk aver-sion and a bequest motive determine a senior?s value function and therefore2Chapter 1. Introductiondrive the senior?s LTCI decision at each period. By carefully constructinghow seniors make insurance and consumption decisions, the chapter is ca-pable of estimating a joint distribution of risk aversion and bequest motivesfrom observed LTCI choices.3Chapter 2Family Name Distribution:The Effect of Social Capitalon Income2.1 IntroductionSocial capital has attracted a great deal of academic attention ever since thework of Putnam, Leonardi, and Nanetti [1993] found a strong correlation be-tween civic engagement and government quality across regions in Italy andconcluded that social capital measured by group membership may spur eco-nomic success. Subsequently, many have explored the connection betweensocial capital and economic performance. For example, Knack and Keefer[1997] found that a one standard deviation increase in social capital as mea-sured by trust level increases economic growth by more than one half of onestandard deviation. But the effect vanishes when social capital is measuredby group membership. Laporta et al. [1997] reported a similar relationshipbetween the social trust level, judicial efficiency and government corruption.Guiso et al. [2004] established that, measured by membership, social capitalhas a strong effect on financial development in Italy. However, Miguel etal. [2005] found no essential relationship between initial social capital andlater industrial development across 274 Indonesian districts where a rich setof social capital metrics were employed.Instead of reporting just another correlation between social capital andeconomic development, this paper examines the causal effect of social capitalon individual economic achievement. By doing so, this paper makes three42.1. Introductioncontributions to the literature. First, by exploiting a cultural institutionunique to Chinese rural society, we provide a unique instrument for socialcapital and thus establish the causal relationship between social capital andindividual earnings. Hitherto, the literature was focused on correlation,rather than causal links. Second, we build a simple model to explain howsocial capital can affect earnings in an economy with imperfect information.Last, our paper presents empirical evidence on different effects of socialcapital between genders: robust and significant effect of social capital onmale peasants but a statistically insignificant effect on females.The deficiency of empirical work on the causal effect of social capitalupon individual economic achievement reflects a number of difficulties. First,social capital is generally measured at the community level. But from anindividual?s perspective, what is the relevant community and with whomdoes she or he form a community? Identifying the effect of social capitalon individual outcomes requires that each member of the community canactually access to the social capital. Thus, a strong community identifica-tion is essential while a segregated community obviously does not meet thiscondition. Second, social capital, whether measured by average trust levelor average club membership of the community members, is almost alwaysconstructed endogenously. Attempting to find a valid instrument is notori-ously hard, in part due to the lack of sufficient theoretical work on socialcapital accumulation. Lastly, individuals often self-select their communitiesbased on community attributes, which can make the endogeneity problemmore severe.We use data from the Chinese General Social Survey(CGSS) 2005 toaddress the above challenges. The CGSS collects very rich information frommore than 4000 family heads, which can be either male or female, from 408Chinese villages. Since a village is an acquaintance society (most villageshave hundreds of years of history) in which villagers know each other sincebirth, almost by definition the residents of a village form a community. Theaccess to village social capital and strong village identification is reinforcedby the fact that migration on economic grounds is very weak because of theresidence permit (hukou) system. During the period studied, nearly the only52.1. Introductionavenue for inter-village migration was through marriage, and even this waslargely confined to women and rarely applied to men. Though rural-urbanmigration is a challenge to the self-selection issue, we find this migration ismostly based on geographic reasons rather than social capital.A unique feature of the Chinese culture permits us to instrument thevillage level social capital by the distribution of family names in a village.Unlike villages in other societies, a Chinese rural village generally only has afew different surnames among its residents. Families with the same surnameare from the same ancestor who first settled in a particular locality hundredsof years ago. At least before 1949, when the People?s Republic of China wasestablished, those families with the same surname were united as a lineage,or family clan, because of both practical needs and the Confucian philosophy.The long history of a village ensures that the distribution of family namesis historically exogenously determined, while the rivalry and cooperationbetween different lineages have a durable impact on social capital.To identify the impact of lineages within a village, we utilize two indicesconstructed from the distribution of family names to measure how rivalryand cooperation affect social capital respectively. The Herfindahl-Hirschmanindex is used to capture rivalry?s lasting influence on social capital, whilethe polarization index is to seize cooperation?s lasting effect. Though inter-village cooperation boosts goodwill, rivalry reduces it. We use these twoindices as instruments. The strength of these two instruments is shownby strong first stage relationships. Two robustness analyses provide moreconfidence in the instrumental variables strategy. To funnel the effect offamily name distribution on social capital, lineages must be built basedon family names. Anthropology research indicates that such link ariseseither from the Confucian philosophy or rice production. A significant firststage relationship between the instrumental variables and social capital isnot found among these villages where neither the Confucius philosophy ispracticed nor rice is cultivated.OLS estimates show a significant correlation between individual incomeand social capital as measured by trust. The correlation is robust to con-trolling for various covariates. A one standard deviation increase in social62.1. Introductioncapital is approximately equivalent to two years of education. However, thissignificant correlation only exists among male villagers.To build the case for causality, we use the instruments to estimate theregression and the IV estimates display a fairly large significant first ordereffect: a one standard deviation increase in social capital is almost equiv-alent to two to four years of education. Similar to the OLS results, theIV estimation finds no significant effect on female villagers. To correct for apotential weak instrument problem, we estimates robust to weak instrument(RWI) following the method proposed by Chernorzhukov and Hansen [2005].The estimated RWI intervals show significant effects at the 95% confidencelevel interval and are robust to various specifications. The RWI estimatesalso support the absence of any significant effects on females.We explored the potential mechanism by which social capital affects theincome of men but not women. To do this, we build a simple model fol-lowing Dixit [2003] and Tabellini [2008] where trades are limited by trustor trustworthiness. We assume trade is the only source of income and ithappens exclusively between different villages. To build the relationship be-tween trust level and income we utilize the result given by Gleaser [2002]that the answer of a subject to the standard attitudinal survey questionsabout trust toward strangers predicts the trustworthiness of the individual.In this setup, social capital affects the volume of trade and therefore incomein the economy. To understand the gender difference, we explore the dif-ference in occupation between male and female villagers. We suspect thatmale villagers? income comes principally from free exchanges with marketswhile females mostly depend on home production. Trustworthiness in a ru-ral economy is crucial for the proper function of markets but has little effecton home production (see Schechter [2007] for an interesting analysis).Our article is closely related to the contributions by Narayan and Prichett[1999] and Miguel and Gugerty [2004]. Narayan and Prichett studied therole of social capital in determining household income in rural Tanzania byconstructing a measure of social capital from a large scale survey data set.They also found a result similar to ours: a one standard deviation increasein social capital increases a household income by at least 20 to 30 percent,72.2. Motivating Theory and Estimation Frameworkequivalent to three years of education. However, their conclusion is vulner-able to the critiques about the validity of the instrumental variables. Theirinstrument choice is built on the assumption that income is not directly in-fluenced by the trust level in strangers or government officials. Miguel andGugerty examined ethnic diversity and local public goods in rural westernKenya and found ethnic diversity is associated with lower primary schoolfunding and worse school facilities. They concluded the inability to imposesocial sanctions in an ethnically diverse community can lead to collectiveaction failures. In this paper, we borrow from their idea about ethnic di-versity but instead focus on the lineage diversity and its impact on socialcapital.This rest of this article is organized as follows. In section 2.2, we presenta simple model to explore why social capital matters and discuss the esti-mation framework. Section 2.3 describes the data and the measurement ofsocial capital. Estimates from OLS are presented as a benchmark analy-sis. Section 2.4 is devoted to building the validity of the instruments. Wefirst give a brief introduction to family names in villages and argue for theexogeneity of the surnames distribution using historical and empirical ev-idences. To prepare for the first stage analysis, we describe in detail theinterplay of rivalry and cooperation of the inter-lineage relationship. Twoindices are constructed to measure different sides of inter-lineage relation-ship. First stage analysis shows a strong relationship between the instru-ments and social capital. Robustness checks provide evidence on the validityof the instruments. Section 2.5 presents our main results and a number ofrobustness checks and section 2.6 provides conclusion.2.2 Motivating Theory and EstimationFramework2.2.1 Motivating TheoryThe mechanism through which social capital affects economic performancehas been extensively examined by recent literature. However, the link82.2. Motivating Theory and Estimation Frameworkfrom social capital to individual economic achievement is rarely explored.Narayan and Prichett [1999] is one of few papers that briefly discusses sev-eral channels through which social capital would affect individual economicachievement.In this subsection, we present a simple model that is designed for ourempirical analysis. Thus, the model only partially explains how social capitalmatters at individual level. While it may not wholly explain the link, webelieve that the mechanism it posits is particularly interesting.The central objective is to show how the trust level of a village de-termines its volume of trade with others and therefore the income of itsresidents since the model assumes income is exclusively from inter-villageexchanges. Bowlus and Sicular [2003] discussed inter-village movement oflabour and concluded the factor market was underdeveloped in rural China.Chen, Mu and Ravallion [2009] discussed the spillover effect of inter-villagetrade. However, this assumption is relatively harmless because the modelcan be expanded to allow income from intra-village production as well. In arecent paper by Guiso, Sapienza and Zingales [2009], using data on bilateraltrust between European countries, these authors found that lower bilateraltrust leads to less economic interaction even after controlling for nationalcharacteristics.Throughout we assume a subject?s trusting level towards strangers rep-resents the level of trustworthiness of the subject. This is justified by theresult in Glaeser et al. [2000] where they found the standard attitudinalsurvey questions about trust predict trustworthy behavior much better thantrusting behavior.Consider the following static model adapted from Dixit [2003] and Tabellini[2008]. Villages indexed by their normalized social capital ki = i ? [0, 1] areuniformly distributed on interval [0, 1]. The uniform distribution assump-tion is just for convenience and our conclusion has no bearing on it. For anyi ? [0, 1], village ki is resided by a continuum of mass 1 villagers. Amongthese villagers, fraction ki are trustworthy denoted as type 1 and the re-maining 1 ? ki are not trustworthy and denoted as type 0. Villagers knowwhich village other villagers are from but not their types. Thus, when two92.2. Motivating Theory and Estimation Frameworkvillagers from two different villages meet, they only know the probabilitythat the other is trustworthy. We assume the villagers? income only comefrom trades with residents of other villages. For example, consider villagersi1 and i2 both from village ki and villagers j1 and j2 both from village kj .This assumption only allows trade between i and j,but not among is oramong js.To complete the model, we make some assumptions regarding to howtrades happen and payoffs from trades. First, villager i from ki matcheswith villager j from kj with equal probability to complete a trade for anyj not equal to i. Since there are infinite villages, the probability is zerofor any specific i and j. Alternatively, We could assume a locally biasedmatching probability like Dixit [2003], but this could needlessly complicateour analysis. Second, once i and j match, they simultaneously choose tradeor not. If any one of them chooses not to trade, each gets payoff zero andthe interaction ends. If both choose to trade, they each get a baseline payoff1 times the type combination multiplier, which is given by the followingmatrix:villager itype 1 type 0villager j type 1 ki + kj , ki + kj ?(ki + kj),ki+kj2type 0 ki+kj2 ,? (ki + kj)ki+kj2 ,ki+kj2where type 1 corresponds to a trustworthy villager and type 0 an untrust-worthy one. Notice the structure of the multiplier is very similar to theto payoff structure of the prisoner?s dilemma. Specially, if both traders aretrustworthy, then each can benefit ki + kj from the trade; if both are nottrustworthy, each benefits ki+kj2 ; finally, if one is trustworthy and the othernot trustworthy, then type 1 loses ki + kj and type 0 getski+kj2 . As in anyprisoner?s dilemma game, we assume throughout that the loss from beingcheated is at least as large as the benefit from cheating.Another feature of the payoff matrix is that the payoff depends on the102.2. Motivating Theory and Estimation Frameworktrustworthiness of ki and kj , which can be justified by the assumption oftransaction cost reduction or a psychological benefit from trade. This featureis crucial for our result because otherwise only the trustworthiness ki wouldaffect the trading strategy.We are especially interested in two questions: Does the expected incomeof an individual from village ki, regardless of the trustworthiness type, in-crease with ki? If so, which type of villager from the same village ki hasa higher income? To answer these questions, we introduce the followingproposition:Proposition. The expected income of villager i from village ki is an in-creasing function of ki, the social capital of the village.Proof. We consider the Bayesian Nash equilibrium. Obviously, the type 0villagers always trade and type 1 villagers trade only if her partner fromkj ? 0.5. The expected income of a type 1 villager is? 10.5(ki + kj)(kj ? (1?kj))dkj = 524 +ki4 for ki ? 0.5 and zero for ki < 0.5. The expected incomeof a type 0 villager is? 0.50ki+kj2 (1 ? kj)dkj +? 10.5ki+kj2 dkj =1148 +716ki forki ? 0.5 and? 10ki+kj2 (1? kj)dkj =512 +ki4 for ki < 0.5. Q.E.D.The above proposition clearly shows a mechanism through which socialcapital can have a positive effect on income. The rest of the paper exploresthis claim. The proof indicates the income of type 0 villager is larger thanthe income of a type 1 villager from the same village. 1 Intuitively, a type 0villager will always be made better off by a trade, and will therefore engagein making transactions whenever encounter with others.2.2.2 Estimation FrameworkAs criticized by Durlauf [2001], empirical work on the causal effect of socialcapital on economic achievement generally suffers from various identifica-tion problems. The problems mainly arise from the lack of a solid theory of1Of course, we can endogenize the payoffs for type 1 and type 0 villagers such thatboth types have the same income in equilibrium. We choose not to do so in order to keepthe model simple.112.2. Motivating Theory and Estimation Frameworksocial capital, e.g., definition ambiguity, measurement confusion and forma-tion and function vagueness (see Arrow [1999] and Solow [1999] respectively).Because of the theory deficit, social capital is often measured through thedecisions made by individuals. In empirical work, the two most prevalentmeasurements of social capital are the trust level and membership status ina community. Without a solid theory, it is difficult to claim that there are nounobservables systematically different between those who trust people andthose who do not. Similarly, communities with higher social capital may besystematically different than others with regard to unobservable attributes.This becomes more serious if individuals can migrate based on social capi-tal. As we will see later, one of the merits of our data is that inter-villagemigration is close to impossible due to the strict residence permit system,hukou, in China. Technically, without the exchangeability 2 assumption,we cannot exclude the presence of unobserved heterogeneity in the sampleunder study.The identification issue is further entangled by reverse causality andneighbour effects. For example, Glaeser et al. [2000] reported a strongpositive effect of education on trust. The human capital theory has long es-tablished the effect of education on income. Using cross country data, Knackand Keefer [1997] showed a strong correlation between economic outcomesand social capitl as measured by trust level. Alesina and Ferrara [2002]reported a robust relation between income and trust level using individualdata from the General Social Survey from 1974 to 1994. Since social capi-tal is measured at the group level, it is necessary to disentangle the socialcapital effect from neighbourhood effects if any. The neighbourhood effectshave long been analyzed by Manski [1997] and Brock and Durlauf [2001].With these considerations, we formulate the following structural equa-2A collection of random variables i is exchangeable if for every finite sample of the ran-dom variables, i1 ...iN , and every permutation operator ?(), ?(i1 ? a1, ..., iN ? aN ) =?(?(i1) ? a1, ..., ?(iN ) ? aN ). See Durlauf [2001] for a good discussion on exchangeability.122.2. Motivating Theory and Estimation Frameworktions:INCOMEi,j = aTRUSTi,j + bSCj + cXi,j + dX?j + eYj + ?i + ?j (2.1a)SCj = ?1Yj + ?2Zj + ?j (2.1b)TRUSTi,j = ?1SCj + ?2Xi,j + ?3INCOMEi,j + ?i + ?j (2.1c)where i and j represent individual i and community (or village in our paper)j, INCOMEij , TRUSTij stands for income of individual ij and the answerto the general trust question reported by individual ij. In the literature ofsocial capital, trust is measured through reports to the question ?Gener-ally speaking, would you say that most people can be trusted or that youcan?t be too careful in dealing with people??. As discussed, the inclusionof TRUSTi,j is necessary for capturing the unobservables to ensure the ex-changeability assumption. SCj is the social capital of village j measured byaveraging TRUSTi,j across i for each j. Xi,j is a vector of individual char-acteristics, X?j is the average of Xi,j across i within village j. The inclusionof X?j is necessary to disentangle the neighbourhood effects. The neighbour-hood is usually defined as X??i,j , the average that excludes the own person?scharacteristics. We choose X?j , which does not exclude the own person?scharacteristics due to sample size consideration. However, main results donot rely on the choice, see Table 2.7. Yj is a vector of village attributes.Zj is the excluded instrument for SCj . The potential endogeneity relationsdepend on the correlation between ?i and ?i, ?j and ?j and ?i and ?j .The equation 2.1a states the relationship between a villager?s income andindividual and village attributes. The equation 2.1b states the relationshipbetween social capital and instruments. This equation implicitly assumesthat social capital is a characteristic of a village on which no single individ-ual has impact. The equation 2.1c states that individual trust decisions arebased on the group trust level (social capital) and individual attributes. No-tice that by taking expectation on both sides and using the iid assumption,we get what Manski [1993] called the endogenous effect.We have two alternative specifications for the estimation framework.First, we can estimate equation 2.1a directly, which has the merit that132.3. Data and OLS Estimatesit captures the potential individual unobservables by including covariateTRUSTi,j and the demerit that it does not consistently estimate the coef-ficient of SCj since from 2.1c it is obvious TRUSTi,j absorbs some effectof SCj . Second, we can use the reduced form as the estimation framework.Combining the equations 2.1a to 2.1c, one therefore arrives at the reducedform expressionINCOMEi,j = ?0SCj + ?1Xi,j + ?2X?j + ?4Yj + ei + ej (2.2)where ?0 is the coefficient of interest. Applying the reduced form to builda causal relation implicitly requires that ?i in equation 2.1c satisfies theexchangeability condition. This can be justified by the setup in the modelthat individual trust decision is based on group attributes. In general, thereduced form is more appropriate since we are mainly focused on the effectof social capital. Thus, in the following regression, we report estimates of?0 by adopting equating 2.2. We also estimate the coefficient of interest byusing equation 2.1a and arrive at similar results.2.3 Data and OLS Estimates2.3.1 DataThe main data set in the paper is the rural part of the Chinese GeneralSocial Survey 2005 (CGSS) conducted by the department of sociology, Ren-min University, China. This data set was collected in October 2005 throughinterviews with households from 401 villages in 75 counties. About ten fam-ilies from each village are randomly picked to take the survey. A householdhead, who can either be the husband or wife, is asked for the householdmembers? information, individual basic information, family background in-formation, value and attitude, and community governance questions. Theindividual?s basic information includes his/her annual income, education,age etc; family background includes his/her parents? education.Among the questions in the value and attitude part is the question like?When it is not directly related with money, to what extent do you think142.3. Data and OLS Estimatesthe following people are trustworthy??. The answers are scaled from 1 to5 where 1 represents ?most people can?t be trusted?, 2 ?many can?t betrusted?, 3 ?half can be trusted and half can?t be trusted?, 4 ?many canbe trusted? and 5 ?most can be trusted?. The ?following people? rangesfrom your neighbours, the villagers with the same surname, classmates, andstrangers etc.. Trust level in strangers is chosen as a measure of social cap-ital. The first reason for this choice is the reliability of this answer. Sincemany of the interviews are conducted under the presence of neighbours orfriends, the answers to friends or neighbours may be biased. Second, ourmotivating model suggests that the mechanism through which social capitalaffects income hinges on the private information of a villager?s trustwor-thiness type, which is most suitable if trust attitude is toward strangers.Third, Platteau [2000] stressed that the generalized codes of good conductlike honest and trustworthiness towards a population at large are essentialfor a modern market economy. Therefore, social capital as measured as trustto strangers can better capture causal effect in a market economy environ-ment. Further, we adopt the mean of the trust answer as the village levelsocial capital following convention. Another measure of social capital, themembership in clubs and associations, is not adopted here because of theextreme low variation in the sample.Figure 2.1 shows village average trust level in different population groups.From left to right, the adjacent line, whiskers and divided box are corre-sponding to the lower adjacent value, 25 percentile, median, 75 percentileand upper adjacent value. It illustrates that trusting is most limited to thosevillagers who have close connections, i.e., people from the same village. Thetrust in strangers is very low: more than half the villages on average say?many strangers can?t be trusted?. Notice the differences between the trustin villagers with the same surname and with a different surname, which issignificant at 1 percent significance level.As Deaton [1997] pointed out, it is difficult to accurately measure theincome of self-employed, especially peasants. The difficulty is partially be-cause of the sensitivity of the topic, which causes the surveyed income tobe underreported or even biased. However, this does not seem to be a se-152.3. Data and OLS EstimatesFigure 2.1: Trust Level in Different Groups162.3. Data and OLS Estimatesrious issue in our data set because in the Chinese rural villages, personalincome is always a public topic among neighbours, friends, and relatives.Family members who are employed in agriculture or family business makesit hard to distinguish personal and family income. It makes the estimate ofincome more difficult. Here, the potential difficulty is the inability to mea-sure opportunity cost. This does not apply to our case since the alternativeof working in town village enterprises(TVEs) can be used as a benchmark.Based on these observations, we claim the individual income in the data setis reliable. To verify it, we compare the reported income in the data set withthe peasants income from Chinese Household Income Project (CHIP) 2002conducted by National Bureau of Statistics, which is regarded as the mostreliable data in income. Following the selection criteria in table 2.1, we findthe average (standard deviation) income reported in the CHIP 2002 dataset is 8.30(.74) for male and 8.29(.66) for female, which are very close tothe values in table 2.1. We also compare the average income using differentcriteria, and find no material differences.The data set also includes some village level information from the vil-lage head, which includes a village?s area, its population, its distance tothe county seat and to the nearest market, its literacy rate and high schoolgraduation number, amount of clinics, schools and TVEs (town village en-terprises) in the village, and landform of the village. One of the specialfeatures of the data set is the inclusion of proportions of families who haveone of the top three surnames in the village. For example, in one of thevillages, the proportions are 40%, 30% and 20%. Thus, in the village 40%of families have the same surname and this surname is the largest one in thevillage, and so on. Notice the total proportion of the top three surnamesis 90% which implies there are 10% of families in the village with familyname(s) other than the top three.During the study period, several features of the Chinese society are es-pecially notable. To control for rural-urban migration, a residence permitsystem, hukou, was first created in the later part of the 1950s and is still ef-fective today. The key part of rural-urban migration under the hukou systemis the hukou conversion process, which basically requires a rural resident,172.3. Data and OLS Estimatese.g., a peasant in a village, to meet both qualification and quota criteria tobe eligible to migrate to an urban area. It is estimated the annual quotaswere extremely small, about 0.01 to 0.02 percent of the rural population ofeach locale. Since 1980s, many rural labours migrated to urban areas thoughworking on urban jobs and residing for the most part in towns and cities3, though they are not legally considered urban workers since they have noaccess to local schools, urban pension plans, public housing and other rightsthat are available to those with urban hukou. This urban-rural migrationcould potentially raise a self-selection issue. However, as pointed out byZhao [1999], more people actually choose not to migrate and the migrationthat does occur is largely circular. Further, such migration is mostly basedon geographic choices rather than social capital considerations (see Zhangand Song [2003]).On the other hand, state control of land closes the channel of rural-ruralmigration. Even today, private land ownership does not exist. Though apeasant is entitled to have usage right of some arable land in his/her birthvillage, the right is not transferred to the peasant if the peasant migrates toother villages. As become clearer with our discussion, one possible threat toour adoption of instrumental variable is that those families with a dominatesurname are assigned more land or more fertile land even if allocated landsize is similar. However, Brandt etal. [2002] analyzed a survey data andfailed to reject the hypothesis that cultivated land is allocated on average indirect proportion to family size. Therefore, the hukou system and absenceof land ownership makes the Chinese village an ideal place to study socialcapital.In this paper, we only consider villagers aged 20 to 55 with an incomegreater than one thousand two hundred Chinese Yuan. The age thresholdis chosen based on the observation that 20 to 55 is regarded as the work agein many rural areas. The income threshold is to eliminate these not activelyparticipate in labour market. Our empirical results are not sensitive to thechoice of the thresholds. Table 2.1 presents the basic descriptive statistics3The number is estimated to be about 190 million in 2009, increasing from about 20million in 1980s. See Chan and Zhang [1999] and Chan [2010].182.3. Data and OLS Estimateson key characteristics. Note that panel B lists the village level averages ofage and educations defined as proportions of the villagers satisfying somethreshold conditions. In a previous version, these variables were defined ascorresponding mathematical averages. In either case, the results are almostidentical.Instead of clustering our data set at the village level, our regressionanalysis clusters at the county level. Many students have pointed out thatthe Chinese peasants mostly limit their social and economic activities withina county. Therefore, villagers? income from the same county are highlycorrelated, which is also reinforced by the fact that the county is the basicadministrative unit. Another practical reason is that clustering at the villagelevel sometimes could not generate enough observations for our regressions.We also perform the analysis clustering at the village level where possibleand find no substantial difference.2.3.2 OLS EstimatesTable 2.2 reports OLS estimates of individual income on social capital andother controls. These results are useful both to show correlation in dataand for comparison to the the IV estimates. Panel A pertains to the es-timates using only male villagers. Column (1) reports the estimate fromthe basic regression without any other controls. The estimate shows thata one unit increase in social capital is correlated with 15% higher income,a relation significant at the 95% confidence level. Column (2) reports theestimate after controlling for individual characteristics including age, educa-tion and parents? education. Column (3) additionally controls for the samecharacteristics across villages, as defined in as defined in 2.1. As discussed,these variables are used to disentangle any neighbour effects. Column (4)additionally controls for village characteristics including the variables listedin Table 2.1. The results in Panel A present a robust correlation betweenincome and social capital. Panel B displays the results for female villagers.In contrast with the estimates in panel A, the results indicate there is nosignificant correlation between income and social capital for female villagers192.3. Data and OLS EstimatesTable 2.1: Descriptive StatisticsValue ObservationsPanel A: Individual CharacteristicsIncome: Male 8.27(.73) 1247Female 8.08(.64) 1125Education: Male 6.97(2.89) 1308Female 5.25(3.35) 1281Age: Male 39.77(9.29) 1308Female 38.55(8.98) 1281Father?s education: Male 2.32(2.79) 1308Female 2.64(3.54) 1281Mother?s education: Male 1.53(1.89) 1308Female 1.80(2.63) 1281Panel B: Average CharacteristicsVillager Aged 0.45(.19) 399Villager Educated 0.56(.22) 399Father Educated 0.08(.10) 399Mother Educated 0.02(.06) 399Panel C: Village CharacteristicsSocial Capital 1.87(.59) 399Population(log) 6.03(.71) 393Area(km2) 7.47(.91) 391Distance to county seat(km) 29.49(21.87) 393Distance to market 4.89(7.20) 393Clinics 2.21(.2.26) 292Primary Schools 1.87(3.61) 295Illiterate(percentage) .21(.33) 390High school graduates .27(.42) 389Number of TVEs 2.96(6.65) 389Notes: Standard deviations are in parentheses. In Panel B, thevillager aged is the proportion of villagers that are older than 45,and the villager educated, father educated and mother educated arethe proportion of villagers, villagers? fathers and villagers? moth-ers, respectively, that have at least 5 years of formal school educa-tion. Social capital is measured as the village average trust level instrangers.202.4. First Stage AnalysisTable 2.2: Effect of Social Capital on Income: OLS Estimates(1) (2) (3) (4)Panel A: MaleSocial Capital .15(.06)** .14(.05)** .14(.05)*** .14(.05)***R2 .01 .13 .15 .19F 6.07 15.21 12.41 7.43Obs. 1247 1156 1134 1113Panel B: FemaleSocial Capital .02(.06) .02(.06) .01(.05) .03(.06)R2 .00 .05 .09 .10F .14 5.71 3.92 5.16Obs. 1125 1034 1031 1001Controls:Individual Charac. YES YES YESAverage Charac. YES YESVillage Charac. YESNotes: Standard errors are in parentheses. A single asterisk denotes significanceat the 10% level, double for 5%, and triple for 1%. The dependent variable isincome.regardless of the specifications. This result is very informative in that itposits a channel to think how social capital affects individual economic out-comes, which is still largely unexplained in academic research. Because ofpotential endogeneitty problems, we cannot conclude that there is evidenceof an causal effect.2.4 First Stage AnalysisBecause of reverse causality and omitted variable problems, OLS estimatesare unlikely to uncover the causal effect of social capital on economicsachievement. To build the case for causality, we outline a source of ex-ogenous variation in social capital with regards to the economic outcometoday. Following the framework, we then formally analyze the first stagerelationship. We examine the exclusion assumption in the next section.212.4. First Stage Analysis2.4.1 Village, Lineage and Family NameThe exogeneity comes from village level variation in family name distri-bution, which is equivalent to the claim that family name distribution ishistorically determined hundreds of years ago and has remained essentiallyunchanged since then. Moreover, the residence permit regulation and onechild policy reinforce the claim.Using historically determined events as excluded instruments is not novelin economics; i.e., Acemoglu and Johnson [2005] applied colonial history toinstrument for two different types of institutions. Adopting distribution asan exogenous variation makes our first stage analysis more subtle since thevalidity of the instruments now relies on appropriate constructions from thedistribution. The exclusion assumption is warranted by the fact that lineagesystem was abolished and replaced with political class by the CommunistParty in 1950s. This subsection presents detailed historical and empiricalevidence to support our claim after a brief introduction to the backgroundof a typical Chinese village, which is mostly adapted from Hu [1983] andBaker [1979].Within the structure of Chinese society, lineage occupied a prominentplace for many centuries until 1949 when the People?s Republic of Chinawas established. A lineage is a corporate group of families which celebratesritual unity and is based on demonstrated descent from a common ancestor(Watson [1965]). Therefore, a lineage traces its ancestry to one ancestorwho first settled in a given locality. The rites in his honor and those forlater ancestors serve as a reminder of kinship bonds. However, women wereexcluded from the lineage since the inheritance of the kinship was predomi-nantly patrilineal.A lineage differs from a family in its scope and functions. A family,which includes parents, children and grand children, is an economic unit tofacilitate child rearing, while a lineage is a patrilineal clan, including all thefamilies with husbands descended from the same distant ancestor. A lineageis a much larger but looser organization compared with a family. Generally,a lineage is cooperative in the sense that member families of a lineage own222.4. First Stage Analysisproperty (usually land) in common. However, this property is for religious,educational and relief purpose rather than a means of livelihood. Wherea lineage constitutes one village, village affairs are managed by its leaders.Furthermore, a lineage is interested in promoting the social standing of itsmembers as their prestige raises the reputation and influence of the group.Since family names pass on down the male line and children take thesurname of their father 4, families from the same lineage share the samesurname as the first ancestor. However, families with the same surname donot necessary belong to the same lineage since they might have a differentpedigree. In our data set, since we can only observe the family name dis-tribution, we must assume those with the same surname are from the samelineage. This is of little consequence as we focus on data from small villages,where a surname almost always implies a lineage. We therefore use familyname and lineage as synonymous in this paper.Most Chinese villages have hundreds of years of history, beginning whenthe first ancestor of a lineage founded the settlement. During that period,others moved to the village and began other lineages. Reliable historicalstatistics are, of course, nonexistent, but support can be drawn from theanthropology literature. Jing [1996] described a typical village in a ruralcounty of Gangsu Province:Interpersonal relations in Dachuan (village) are tightly knit bydescent and marriage. Although it is not a single-surname vil-lage, 85 percent of the local households in 1992 were surnamedKong. The balance comprised 16 other surnames. Except fora group surnamed Li, whose ancestors had settled in Dachuanearlier than the Kongs, the others come to the village as refugeesfrom war and famine.These Kongs trace their ancestry to Confucius through a Guang-dong born ancestor who migrated to Gansu and settled in (this4It only happens under extremely rare cases that a boy is surnamed with his mother?sfamily name. Actually, the author never observed it in his own village, Liji Chun at AnyiCounty, with a population around 400.232.4. First Stage Analysisvillage) six centuries ago...the Kongs had long identified them-selves with Confucian heritage by three means: a carefully guardedcollection of genealogical records, a cycle of ancestral rituals anda temple dedicated to Confucius.The above citation indicates that kinship relation still plays an importantrole in today?s Chinese villages despite the fact that lineage in the traditionalsense of ownership of common assets had not existed since the establishmentof the People?s Republic of China. Note that the first ancestor of Kong?slineage moved to the village almost six hundreds years ago after lineage Li?sancestor, and the ancestors of families with different surnames sequentiallymoved to the village because of war and famine.With this evidence, we conclude that the lineage or family name dis-tribution at the village level is determined historically. We next show thatcontemporary economic conditions do not influence the distribution. With-out time series data on it, this cannot be shown effectively. Alternatively,the claimed exogeneity is built by showing that families from different lin-eages or income groups have the same amount of boys. The focus on theamount of boys is justified by the fact that only boys usually stay in thevillage when they grow up and thus are considered as part of a lineage.Furthermore, the strictly regulated residence permit hukou policy reinforcesthe claim since villagers cannot freely migrate between different villages.One speculation that could potentially invalidate the exogeneity assump-tion is that richer families in a village could have more boys than otherfamilies. If this is true, the lineage distribution would depend on economicstatus. To test this hypothesis, we use the data set China Household IncomeProject(CHIP) 2002, which includes detailed information on family memberand household income. The coefficients(standard error) of the regressionof the number of boys on household/head income are 0.05(0.13)/0.06(0.13).These are not significant. The conventional wisdom that wealthier familiescan have more children or boys does not apply for two reasons. The one childpolicy makes it very expensive to have more children, especially more boys(Ebenstein [2008]). The prevalent, strong son preference in rural Chinese242.4. First Stage Analysisvillages provides incentives for a poor family to have a boy by all means.The Confucian doctrine of filial piety teaches ?there are three ways to beunfilial, the worst is not to produce male offspring?. A traditional Chinesefamily would employ all of its assets to have a son. This is verified by thefinding that wealthier families do have more children but not more boys.The second conjecture is migration between different villages as a re-sponse to economic circumstances. However, this is not warranted sincelineage is based on descent relationships and migration definitely cuts thisrelationships without much benefit (see more on inter lineage relationshipsin the next section). The strict residence regulation policy, hukou, since1950s closes the door for any migrations on grounds of personal economicconsiderations (Chan and Zhang [1999]).In summary, the historical and empirical evidence reasonably supportsour claim that surname distribution is exogenous with regard to currenteconomic conditions. The last subsections build the first stage relationshipby presenting the inter-lineage rivalry and cooperation relationship, whichare essential for constructing the instruments. During the discussion, theinter-lineage relations are limited to the lineages from the same village. Abetter viewpoint would be the inter-lineage relations among neighbour vil-lages. The latter one is infeasible in our case because of the data limitation.2.4.2 Inter-Lineage Rivalry and CooperationUnderstanding interactions between lineages is fundamental to understand-ing and capturing the effect of lineage composition on trusting behaviourat a village level. In this section, we briefly introduce some findings fromanthropology about how lineages battle with each other and cooperate as aunit when it is necessary. The assumption is while serious rivalry impedesdevelopment of a trust, cooperation between different lineages builds it.How rivalry affects trust has a long history of debate among social sci-entists. Intuitively, a fierce competitive market impedes reputation building5 and deters trusting attitude. From a cultural evolution viewpoint, rivalry5See Bowles [1998] for a comprehensive review.252.4. First Stage Analysiscan influence the structure of social interaction among lineages, thereforeaffecting social norms of parent-to-son trust transmission process(see Bisinand Verdier [2001]; Francois and Zabojnik [2001]). Though a positive re-lationship between trust level and competitiveness was reported in someresearch(e.g., Francois and van Ypersele [2009]), rivalry differs from compe-tition and does not foster a trusting attitude. Rivalry is more stressful andaggressive. And unlike competitors, rivals follow no rules.Lineages vie not only for land or water, but sometimes prestige and feng-shui, which is believed to bring fortune, wealth, and health to the wholelineage. Beattie [1979] showed that rivalry among lineages in Tongcheng,Anhui province, for social prestige is a critical element for the increasingsuccess of the lineages in the examination system. Baker [1979] noticedhow the Confucianism doctrine of filial piety turns ?family by being inward-turned . . . outwardly aggressive.? and ?The same applied to the lineage: butthe lineage was also expansionist.? Baker [1968] also illustrated a lineagebattle with other lineages for good feng-shui by tearing down a pagodaafter incorrectly building it:It was thought that the open mouth of the eagle was swallowingsome of the good fortune of the Liaos, so that male children andexamination success in particular were being denied the lineage,and a geomancer suggested that between the village and the eaglea feng-shui pagoda should be built to represent a bird table, thusprotecting the Liaos from the bird?s appetite. Unfortunately thepagoda was not built on the correct site, and other villages tothe west benefited instead, gaining unprecedented successes inexaminations. The lineage tore down the pagoda...The rivalry for land, water, and sometimes control over a local marketeven results in feuds. Freedman [1966] vividly described some feuds amonglineages and concluded ?...that fight between lineages... was an importantcharacteristic of social life...? Hu [1983] gave a detailed description abouthow two lineages fought for a small mountain with woods, which finallyclaimed seven lives. Though loyalty encourages taking part in the fighting262.4. First Stage Analysisin a full scale feud, the rewards are provided by the lineage, which includestaking care of the families, widows, and immortalization ritual to those deadin the war.However, some students (see Strauch [1983]; Johnson [1976]) observedthat patrilineal ideology does exert an undeniable influence on inter-lineagerelationships, but it does not mean a village compound of different lineagescannot work together in times of need. It seems reasonable to assume thatlineage solidarity can coexist harmoniously with community solidarity froman infinitely repeated game perspective. The field work by Strauch [1983] ina multi-lineage village named FungYuen supports this point. In the village,a sense of village membership is at least as strong as the lineage identificationand village exogamy is strictly enforced. Johnson concluded that:The people ...lived in an immediate social world populated chieflyby themselves, and they developed patterns of cooperation ineconomics and ritual spheres that united them even more stronglythan their separate ancestral loyalties divided them.There are many reasons for strong cooperation rather than rivalry invillage life. Risk sharing or cost sharing are among the most prevalent. Forexample, lineages in the same village would unite to compete with othervillages for a good tutor for children. To defeat robbers and pirates, familiesfrom different lineages come together. Strauch [1983] reported how peo-ple from different lineages formed groups and shared resources during theJapanese occupation period in the village. Baker [1966] detailed how lin-eages in Hong Kong draw together and cooperate after being attacked bypirates:... but rather lineages of the five clans which came togetherand each purchased a share in the temple.... Not only was landpurchased and a temple built with this money, but also a ferryboat was bought to assist all members of the five clans to crossthe Sham Chun River to get to the large market town of ShamChun, with which all had dealings. The share-holding lineages272.4. First Stage Analysistook part in an annual feast at which the business of the templewas discussed, the feast being paid for out of temple funds.From the above discussion, it seems safe to conclude that rivalry andcooperation among lineages are not mutually exclusive in village life. Thecoexistence of multifaceted social interaction strategies makes it unclear howthe lineage composition affects social capital. Thus, it is necessary to usedifferent methods to capture the effect of rivalry and cooperation on socialcapital. For this reason, we next discuss how to construct two instrumentsto respectively capture rivalry and cooperation among lineages.2.4.3 MeasurementThis section constructs instrumental variables from a village?s lineage com-position. The key idea is to capture the two sides of the inter-lineage rela-tions by proposing two different measurements. Concretely, we employ thefractionalization or Herfindahl-Hirschman index to capture rivalry and thepolarization index to capture cooperation. The polarization index, thoughdesigned to capture total social antagonism of a polarized society, measuresmutual affection from cooperation quite well. Let us denote the lineage dis-tribution in a village as (pi,1) = (pi1,11; ...;pin, 1n), where the positive integern is total lineages in a village, pii is household or population proportion ofthe lineage i in the village, which satisfies?ni=1 pii = 1, and 1i is an indexfor lineage i.Before presenting the formal analysis, Table 2.3 illustrates the associa-tion between social capital and family name composition. Columns 1 and2 show the OLS estimates of social capital on the share of the most com-mon surname while columns 3 and 4 list the results on the aggregated shareof surnames other than the top three. The relationship is rough, but thegreater the prevalence of the most common surname, the more inclined to-ward higher social capital. In contrast, the greater of the percentage ofsurname other than the top three, the more likely toward lower social capi-tal. Though the exact channel of the association is not yet clear, it revealsthe basic link between the surname composition and social capital.282.4. First Stage AnalysisTable 2.3: Families Name Composition and Social Capital(1) (2) (3) (4)Family Name Percentage .54(.17)*** .61(.17)*** -.004(.001)** -.005(.001)**R2 .05 .10 .04 .08Obs. 395 393 358 350Village Charac. YES YESNotes: The dependent variable is social capital as measured by trusting attitude. In columns1 and 2, family name percentage is the the percentage of the most common surname, while incolumns 3 and 4 is the share of the surnames other than the top three. The standard errorsare in parentheses. A single asterisk denotes significance at the 10% level, double for 5%, andtriple for 1%.The fractionalization index (FI), defined as?ni=1 pii(1 ? pii) = 1 ??ni=1 pii2 , has been widely adopted in conflict research to examine the rela-tionship between ethnic diversity and poor economic performance resultingfrom investment deterrence, impediments to technology innovation, or po-tential conflict (see Easterly and Levine [1997]; Alesina et al. [2003]). Theindex is commonly interpreted as the probability of two randomly selectedindividuals will not be of the same group. Alesina and Farrara [2002] re-ported a significant negative correlation between the ethnic fractionalizationindex and trust levels. Vigdor [2002] gave a simple behavioral model to pro-vide an economic motivation for fractionalization effects.To capture the cooperation effect, we use the polarization index (PI) de-veloped by Esteban and Ray [1994] and Montalvo and Reynal-Querol [2002].By assuming that a population of individuals may cluster according to somecharacteristics and clusters are mutually antagonistic, the polarization indexwas originally designed to measure polarization through total social antago-nism. Since we are trying to measure how cooperation affects trust, insteadof assuming antagonism amid different lineages, we assume that cooperationamong lineages brings mutual affection. This is appropriate considering so-cial capital is measured by the average trust level in strangers in our dataset. Thus, our polarization index measures the total inter-lineage affectionin a village.The construction follows Esteban and Ray [2004], except we change292.4. First Stage Analysisantagonism to friendship. Assume that intra-lineage identification weak-ens inter-lineage friendship and inter-lineage cooperation accentuates af-fectionate attitude in the village. Specifically, let the identification func-tion I : [0, 1] ? R and cooperation function c(d(1i, 1j)) be an increasingfunction with properties c(0) = 0, where d(1i, 1j) is the social distancebetween two lineages i and j. To capture the affection that any individ-ual from lineage i feels towards lineage j, we define the effective friend-ship function F (I, c) to be strictly increasing in c. The total polariza-tion in a village level is defined as the sum of all the effective friendship:P (pi,1) =?ni=1?nj=1 piipijF (I(pii), c(d(1i, 1j))). Under some reasonableaxioms, it can be shown that the polarization index can be expressed asP (pi,1 : ?) = K?ni=1?nj=1 pi1+?i pijd(1i, 1j), where K > 0 and ? ? (0, 1.6]is the degree of polarization sensitivity. Following Montalvo and Reynal-Querol [2005], we further substitute the social distance d(1i, 1j) = 1 if i 6= jand d(1i, 1j) = 0 otherwise, and the sensitivity parameter ? = 1. Thisassumption still allows for intra-lineage friendship. It is maintained be-cause we want to measure the effect of inter-lineage cooperation on socialcapital. Finally, our polarization index is PI = 4 ??ni=1 pi2i (1 ? pii) =1??ni=1(0.5?pii.05 )2pii 6.The CGSS data set only includes the top three lineages? proportion foreach village, thus some of the distribution is unobservable, truncated beyondthe top three lineages. The total share of the top three lineages is lessthan 85 percent for two thirds of the villages in the data set. Ignoringthe data truncation can introduce serious bias. We take two steps to dealwith this problem. First, for the villages truncated beyond the third biggestlineage, we assign the residual portion to other lineages equally; the numberof unobserved lineages is calculated based on the shares of the third largestlineage. For example, consider a village with the lineage composition (pi) =(pi1 = 40%, pi2 = 20%, pi3 = 15%). The sum of the top three lineage is40 + 20 + 15 = 75 percent. The residual share is 100? 75 = 25 percent. The6Actually, ? = 1 is the only level generating the polarization index that satisfies somegood properties of polarization and K = 4 is the only value such that the index rangesfrom zero to one. See Montalvo and Reynal-Querol [2002] for detailed proofs.302.4. First Stage Analysisnumber of the non-observed lineages is calculated by the ceiling integral of25/15 + 1 which is 3. So, the residual share is equally divided to the fourth,fifth and sixth lineages and each lineage has a share of 25/3 = 8.34. Thesecond step is introduced in section 2.5.This method exploits the rank information included in the data set. It re-lies on the assumptions about the unobserved lineages. However, comparedwith other methods that rely on distributional assumptions, we believe itis more acceptable since it makes the least assumptions. Different methodsthat assume unequal distribution among the left proportion were tried aswell, but no differences were found and thus these results are not reportedhere.Figure 2.2 presents the relationship between the FI capturing rivalryand PI capturing cooperation. Obviously, FI has a horizontal U-shapedrelationship with PI. In other words, PI maximizes when FI is around 0.5,and minimizes when FI is around 0 or 1. A FI around 0 or 1 implies a villageconsisting of either one dominating lineage or many small lineages. Undereither situation, rivalry force will be taken over by a need for cooperation.Similarly, a FI around .05 must be from villages with several dominatinglineages, a situation freezing cooperation.2.4.4 First Stage AnalysisThe PI index captures cooperation and FI index measures rivalry. Thus,social capital as measured as trusting attitude should have a positive corre-lation with PI and a negative one with FI.Figure 2.3 plots the partial correlation between social capital and theconstructed instruments at the village level, where the upper panel is forthe rivalry index and the lower for the cooperation index. Notice a similarfigure at the individual level should have the same pattern except a tighterconfidence interval for the fitted line and thus is omitted. The visual repre-sentation shows a strong first stage relationship between social capital andinstruments: a negative partial correlation for the FI and a positive onefor the PI as expected. The higher the FI, the more lineages concentrated,312.4. First Stage AnalysisFigure 2.2: Fractionalization vs. Polarization IndexFigure 2.3: Partial Correlation of Social Capital and Instruments322.4. First Stage Analysislead to more rivalry. Cooperation is reinforced in a more ?polarized? village,as suggested by a higher value of PI. The significant correlation adds con-fidence in the validity of the instruments to capture rivalry and cooperationamong lineages.Table 4.1 details the regressions from the first stage at village level. PanelA pertains to the FI and panel B to the PI. Two features deserve specialattention. First, the regression verifies the claim that the constructed in-struments have a significant effect on social capital subject to controllingvarious covariates. Technically speaking, what is relevant to the analysis isthe individual level first stages. The village level can be treated as a ro-bustness check since it is possible the strong individual correlation betweenthe instruments and social capital might mainly result from repeated ob-servations. The estimates in the table eliminate such concern. Individuallevel results are available and support the above analysis. Second, a strikingfeature in the table is that F?values in the first stage regressions are lowerthan what is needed for a strong instrument, say an F?value of 10 suggestedby Stock, Wright and Yogo [2005]. This is true even in the individual levelregression. Thus, the weak instrument problem is a concern and should betreated carefully.Overall, Figure 2.3 and Table 4.1 show that there are significant firststages for social capital at the village level, but the first stages potentiallysuffer from the weak instrument problem.2.4.5 First Stage RobustnessThe first stages show our constructed instruments have a significant rela-tionship with social capital as expected. Nevertheless, our measurement ofsocial capital is based on the decisions made by subjects (to what extentto trust strangers). It is possible what measured is not social capital, butsomething unobservable in the data that affects the decisions. To addressthis concern, we perform two robustness tests based on the origins of lineageand anthropological evidence.The connection between family name and lineage is mainly based on332.4. First Stage AnalysisTable 2.4: First Stage Regression for Social Capital(1) (2) (3)Panel A: Fractionalization IndexFractionalization Index -.50(.19)** -.52(.19)** -.57(.19)***Controls:Average Charac. YES YESVillage Charac. YESR2 .034 .040 .013F?value 6.75 1.73 3.60Observations 378 375 344Panel B: Polarization IndexPolarization Index .28(.17)* .28(.17)* .31(.17)*Controls:Average Charac. YES YESVillage Charac. YESR2 .010 .013 .099F?value 2.83 .81 3.03Observations 378 375 344Notes: Standard errors are in parentheses. A single asterisk denotessignificance at the 10% level, double for 5%, and triple for 1%. Thedependent variable is social capital measured by trust in strangers atvillage level.342.4. First Stage AnalysisConfucian philosophy, so is the first stage relation. It is standard to assumeConfucianism provides the core values and norms of Chinese society, butthese values are actually practiced mostly by Han Chinese. It is questionableto what extent Confucian values can be generalized to other ethic groups.Shein [1994] illustrated how a member of the Miao Chinese is so differentfrom the Han Chinese in terms of culture, including ritual activities andspiritual beliefs etc.. Thus, our first test is to divide villages into two types;those that only include Han Chinese and those populated with all otherChinese. To simply the analysis, any village with at least one minoritypeasant is treated as a minority village. In total, 59 villages out of 318 arecategorized as minority villages.Secondly, anthropologists like Freedman [1966] have argued the agri-cultural surplus associated with a rice economy initially made possible theestablishment of corporate estates which in turn promoted the developmentof large patrilineal communities. He noted a correspondence between re-gions with large, localized lineages and areas of rice cultivation. Pasternak[1969] extended this claim and argued that the exigencies of frontier lifewere crucial for the emergence of strong lineages. There is, in addition tothe cooperative effort necessary to bring wild land under cultivation, a needfor organized defense against bandits invaders. Rice cultivation and frontierlife in Chinese history were generally a phenomena of South China. Fromthis evidence, it seems reasonable to conclude that lineages are more stableand effective in south China. Our second test differentiates villages formSouth and non-South, including North and West China 7.Table 2.5 and Table 2.6 present the estimates. The dependent variableFractionalization Index*Minority is the interaction item of FractionaliztionIndex and Minority dummy variable. Similarly for other interaction items.Both panel A and panel B show that in the minority villages where thelineage system is very weak or nonexistent, the interactions have no essentialeffect on social capital. They also give a significant and robust first stage7By South China, we refer to these provinces: Jiangsu, Zhejiang, Anhui, Jiangxi,Shangdong, Fujian, Guangdong, Guangxi, Hubei and Hunan. All other provinces in ourdata are treated as non-South China.352.4. First Stage AnalysisTable 2.5: First Stage Estimates: Minority Villages vs. Han Villages(1) (2) (3)Panel A: Fractionalization IndexFractionalization Index*Minority -.10(.45) .-.08(.40) -.15(.70)Fractionalization Index -.51(.20)** -.54(.22)** -.47(.17)**Controls:Average Charac. YES YESVillage Charac. YESR2 .25 .28 .40Observations 378 354 344Panel B: Polarization IndexPolarization Index*Minority .07(.44) .15(.38) -.10(.47)Polarization Index .32(.17)* .42(.17)** .40(.16)**Controls:Average Charac. YES YESVillage Charac. YESR2 .20 .23 .34Observations 378 354 344Notes: Standard errors are in parentheses. A single asterix denotes significance atthe 10% level, double for 5%, and triple for 1%. The dependent variable is socialcapital measured by trust in strangers at village level.relationship for those villages consisting of both Han and minority Chinesewhere the lineage system averagely has a tradition and is rooted into dailylife and spiritual beliefs 8.Table 2.6 has a similar structure. The interaction items are the con-structed instrumental variables and the South or North dummy variable.The similar pattern of the estimates brings us more confidence in the in-struments. In both cases, our results negate the possibility that the firststages presented in the last section are driven by other unobservables.These results therefore indicate that the relationship in the first stage arenot likely driven by unobservable variables, which gives us more confidencein using the constructed instruments to investigate the effect of social capitalon individual economic achievement.8This conclusion is even more apparent when we divide the sample into North andSouth (Han vs. Minority) groups and compare the estimates from these groups.362.4. First Stage AnalysisTable 2.6: First Stage Estimates: South vs. Non-south(1) (2) (3)Panel A: Fractionalization IndexFractionalization Index *North .04(.27) -.03(.26) -.10(.18)Fractionalization Index -.66(.23)*** -.65(.25)** -.71(.23)***Controls:Average Charac. YES YESVillage Charac. YESR2 .23 .25 .29Observations 378 378 344Panel B: Polarization IndexPolarization Index*North -.07(.28) -.04(.36) -.03(.25)Polarization Index .44(.21)* .43(.18)** .39(.20)*Controls:Average Charac. YES YESVillage Charac. YESR2 .20 .25 .27Observations 378 378 344Notes: Standard errors are in parentheses. A single asterix denotes significance atthe 10% level, double for 5%, and triple for 1%. The dependent variable is socialcapital measured by trust in strangers at village level.372.4. First Stage AnalysisFirst Stage Mechanism So far, our first stage analysis has shown ev-idence for the strong relationship between social capital and inter-lineagesrelationship. Since social capital is measured as a trust level in strangers, thestrong relationship demands an explore why the inter-lineages relationshipthat most likely happens among acquaintance could affect villagers? trustingattitude towards strangers.Inter-lineage relationship, where villagers and their offspring can be mod-eled as playing an infinitely repeated game, affects players? prior that anopponent is cooperative rather than fully rational. In an experimental envi-ronment to examine how exogenously determined past relationship lengthsaffect trust and trustworthiness, Engle-Warnick and Slonim [2003] foundthat there is significantly more trust after long relationships. This is espe-cially relevant in the Chinese rural villages where inter-lineage rivalry andcooperation dominated the rural social life in history.The diffuseness of trust in neighbours to outsiders is more likely a resultof multi factors, though reputation can be of more effective. Manapata,Nowaka and Randa [2012] discussed in the traditional trust game 9 how alittle information on the trustee?s reputation can lead to robust evolution oftrust and trustworthiness through partner choice. In our model, with theacceptable assumption that a stranger (investors) have some knowledge ofa trustworthiness of a villager (trustee), the result can be applied to explainthe diffuseness of trust.To test the general idea of free market exchange that might boost trust instrangers and general public, we decompose villager?s income into two parts:one from agriculture production and the other one from non-agriculturalmarket activity (including small business and non-agricultural labour in-come). Define market intensity as the ration of non-agricultural income toagricultural income and we find the correlation between market intensity andindividually reported trust in strangers is around 0.43. This result seems9A trust game consists of an investor and a trustee. The investor, endowed with somemoney, makes the first move by either keeping the money or transferring it to the trustee.The value created by interactions based on trust is the money multiplied by a factor? > 1. Finally, the trustee chooses how much to return to the investor and how much tobe retained.382.5. IV Estimatessupport that market exchange might help to diffuse inward trust towardoutsiders.2.5 IV Estimates2.5.1 Main ResultsWe previously mentioned that we need a second step to compensate for thetruncated observation problem. The first step is used to impute the missingvalues which are essential to construct instruments. Its disadvantage is thatthe procedure is vulnerable to interference from noise, as we observed thatamong two thirds of villages the top three lineages are less than 85% of thepopulation. The noise prevents the instruments from possessing significantvariation in the instrumented social capital, making it hard to capture theeffect of interest. To overcome this problem, we generate eleven dummies bythe total proportion of the top three surnames and multiply these dummiesby FI (PI). The first dummy is valued 1 if the total share of the top threesurnames is from 1 percent to 9 percent. Others are defined similarly untilthe last one which is valued 1 if the total share is 100 percent. For example,consider the dummy tenth which equals 1 if a village?s total percentage ofthe top three ranges from 90 percent to 99 percent and 0 otherwise. Wegenerate the dummies according to the sum of the top three because thenoise comes from the imputed values of the unobservable surnames. Thishinges on the sum of the top three and the value of the third lineage. Sinceinstruments are more than one, to get efficient estimates, we adopt two stepGMM estimation. It is more efficient than two stage least squares since theformer can capture heteroscedasticity.Table 2.7 reports the results for males and females respectively. Panel Auses the FI as the instrument while panel B uses the PI. Column (1) doesnot include any controls. Column (2) only includes controls for individualcharacteristics like age, square of age, education and parents? education.Column (3) additionally includes average individual characteristics such asvillage average education and village average age and its square. Column392.5. IV Estimates(4) includes other village characteristics listed in table 2.1. In column (1)of both panels, the coefficients for males are around 0.50 and statisticallysignificant at 10% level. The coefficients for females are neither stable norsignificant. Column (2) to column (4) illustrates this first order effect bycontrolling various individual and village characteristics, the estimates rangefrom 0.34 to 0.56. The standard deviation of social capital is about 0.57 inthe data set, which means an increase in one standard deviation in socialcapital can increase the villagers? incomes by 19% to 32%. Column (5) liststhe estimates with the average village characteristics that do not include theown-person characteristics. Generally, the return of education is about 8%to 10%. So, a one standard deviation increase in social capital was found tobe equivalent to about two to four years of education. As we found in theOLS estimates, the IV estimates also indicate the absence of any effect onfemale villagers.Table 2.7 also reports F values from the first stage regression. TheF values are lower than the conventional recommended value of 10 for anon weak instrument. This implies a very low value of the concentrationparameter, that is the proportion of the social capital?s variation explainedby the included variables. Actually, we have encountered this problem in thefirst stage analysis earlier. As discussed by Stock, Wright and Yogo [2002]and Stock and Yogo [2005] among many others, the estimates can be biasedand the standard inferences based on normal distribution can be misleading.Thus, caution should be applied in reading the results in the table 10.Overall, the results in this subsection suggest that social capital has afirst order effect on income for males, but not for females. Holding all otherfactors constant, a male villager can increase his income more than 19%to 32% if he is from a village with one standard deviation higher socialcapital, which is averagely equivalent to 3 years of education. This result isrobust in the sense that it holds under various specifications. However, thefirst stage F values also imply a potential weak instruments problem, whichcould contaminates the conclusion.10Another potential issue is the small sample bias for IV. As Ham, Kagel and Lehrer[2005] shows, a sample size of 3000 is necessary for a good inference.402.5. IV EstimatesTable 2.7: Effect of Social Capital on Income: IV Estimates(1) (2) (3) (4) (5)Panel A: Instrument Variables: Fractionalization IndicesMaleSocial Capital .50(.28)* .56(.25)** .57(.25)** .35(.17)** .37(.16)**R2 .01 .03 .04 .14 .15F ? V alue 2.10 2.00 1.72 2.02 2.34Obs. 1247 1156 1134 1113 1113FemaleSocial Capital -.08(.18) .03(.16) .03(.16) .01(.17) 0.02(.13)R2 .01 .05 .09 .10 .12F ? V alue 1.25 1.19 1.49 1.61 1.73Obs. 1125 1034 1031 1001 1001Panel B: Instrument Variables: Polarization IndicesMaleSocial Capital .49(.26)* .49(.23)** .50(.22)** .36(.18)** .45(.19)**R2 . 01 .06 .07 .14 .16F ? V alue 2.17 1.94 1.68 1.94 2.31Obs. 1247 1156 1134 1113 1113FemaleSocial Capital .23(.25) .20(.16) .16(.17) .13(.17) .15(.23)R2 .01 .02 .08 .09 0.09F ? V alue .99 1.12 1.51 1.53 1.57Obs. 1125 1034 1031 1001 1001Controls:Individual Charac. YES YES YES YESAverage Charac. YES YES YESVillage Charac. YES YESNotes: Standard errors are in parentheses A single asterix denotes significance at the10% level, double for 5%, and triple for 1%. The dependent variable is individual income.F-Value is the F-value at the first stage regression.412.5. IV Estimates2.5.2 Robustness to Weak IVIn the literature, many methods have been proposed to correct the bias ofweak instruments. However, most of these methods focus on the case ofi.i.d. errors. In this paper, we apply a simple robust inference procedurefollowing Chernozhukov and Hansen [2008]. Our main concern is whetherthe coefficient on social capital is some given values. Under the null that thecoefficient is equal to a given value, the exclusion restriction implies that thecoefficients on the excluded instruments in the reduced form should equalzero. Thus, testing the hypothesis that the coefficients of all the excludedinstruments are zero is equivalent to a test of whether the coefficient ofsocial capital equals the given value 11. Chernozhukov and Hansen [2005]showed that the conventional Wald statistics for testing this hypothesis isasymptotically distributed as a ?2 regardless of the strength of the excludedinstruments. To find the weak IV robust interval, we repeat the above pro-cedure for different values of the coefficient of the social capital. Concretely,we consider the coefficient intervals of [?1, 1] with step 0.001. The meritof this procedure is to avoid the i.i.d. assumption required by many othermethods. Notice we have more instruments than the endogenous variables.The constructed Wald statistics used to test the hypothesis also tests thespecification.Table 2.8 presents the weak IV robust intervals for the coefficient of so-cial capital at the 95% confidence level. As a comparison, we also replicatethe estimates in Table 2.7 and its 95% confidence interval. Panel A givesthe results using the FI as the instruments. For males, the weak IV robustinterval at the 95% confidence level ranges from 0.025 to 0.603. The esti-mated intervals robust to weak IV are generally tighter than the intervalconstructed the conventional way. Also, the two step estimates are closeto the maximum value of the weak robust interval. For female villagers,11To make it clear, consider the following structural model:Y = SC?1 + X?2 + ? andSC = Z? + V , where Y is the income, SC represents social capital, Z is the excludedinstruments and X is the included instruments. The test of ?1 = ?01 is equivalent to thetest ? = 0 under the reduced form Y ?SC?01 = Z?+X?2 +?. Notice here no informationabout the relationship between SC and Z is ever used.422.5. IV EstimatesTable 2.8: Effect of Social Capital on Income: Robust to Weak IV(1) (2) (3) (4)Panel A: Instrument Variables: Fractionalization IndicesMaleSocial Capital .50(.28)* .56(.25)** .57(.25)** .35(.17)**Asymptotic Interval (-.049,1.054) (.045,1.071) (.083,1.054) (.003,.684)Weak IV Robust Interval (.025,.506) (.235,.603) (.254,.555) (.129,.498)Obs. 1247 1156 1134 1113FemaleSocial Capital -.08(.18) .03(.16) .03(.16) .01(.17)Asymptotic Interval (-.437,.273) (-.278,.336) (-.288,.343) (-.323,.332)Weak IV Robust Interval (-.999,1.001) (-.999,1.001) (-.999,1.001) (-.999,1.001)Obs. 1125 1034 1031 1001Panel B: Instrument Variables: Polarization IndicesMaleSocial Capital .49(.26)* .49(.23)** .50(.22)** .36(.18)**Asymptotic Interval (-.017,.1.007) (.053,.943) (.095,.941) (.016,.676)Weak IV Robust Interval (.210,.401) (.283,.644) (.274,.549) (.095,.472)Obs. 1247 1156 1134 1113FemaleSocial Capital .23(.25) .20(.16) .16(.16) .13(.17)Asymptotic Interval (-.208,.455) (-.112,.531) (-.158,.487) (-.205,.475)Weak IV Robust Interval (-.999,1.001) (-.999,1.001) (-.999,1.001) (-.999,1.001)Obs. 1125 1034 1031 1001Controls:Individual Charac. YES YES YESAverage Charac. YES YESVillage Charac. YESNote: This table replicates the Table 2.7. The asymptotic interval is the 95% confidence intervalusing the usual asymptotic approximation. The weak IV robustness interval reports the 95%confidence interval using the weak-instrument robust statistics.432.5. IV Estimatesthe weak robust interval ranges form ?0.999 to 1.001, the whole consideredinterval. These results establish the gender differentiation in social capital.Panel B displays the results where PI are used as instruments. The weakrobust interval for male villagers ranges from 0.095 to 0.644, very close theinterval found earlier. The female results are also unchanged.Thus, the weak IV robust intervals verify our finding from two stepIV estimates. It reinforces the claim that social capital has a significantand robust first order effect on males, but no detectable effect on females.The effect on male villagers varies from 2.5% to 64%, depending on thespecifications. The interval becomes much tighter, from 9.5% to 49.5%, incolumn (4) where all related factors are controlled. Under the assumptionthat social capital has no direct effects on these factors, this interval impliesthat a one standard deviation unit increase in social capital is equivalent toone to three years? schooling.2.5.3 Robustness to Too Much NoiseAt the beginning of this section, we discussed why and how to generateeleven indices to capture variation in social capital. The general IV es-timates and the estimates robust to weak IV show a significant first ordereffect. However, another concern is that the estimates we got in last the sub-sections are mainly driven by the eleven dummies, not by the instrumentsor alternatively, the estimates are mostly driven from the use of too manyinstruments. The implicit argument for the latter concern is that given somany instruments with so much noise, it is possible to get an illusionaryfirst order effect.The first argument can be tested by using only the eleven dummies inthe IV estimation. The regression results show that without multiplyingwith the original instruments, the coefficients for both male and female areinsignificant under various sets of controls.To assess the second argument, we use the following randomization test.We first collect all the villages? family name distributions and then randomlymatch a village with one of these distributions. With this randomly gener-442.5. IV EstimatesTable 2.9: Effect of Social Capital on Income: Randomization Test(1) (2) (3) (4)Panel A: Instrument Variables: Fractionalization IndicesMaleSocial Capital 008(.445) .007(.476) .002(.467) .014(.356)(min,max) (-2.302,1.350) (-1.555,1.282) (-1.345,1.456) (-1.387,.989)Panel B: Instrument Variables: Polarization IndicesMaleSocial Capital -.016(.440) .011(.469) .035(.454) -.007(.342)(min,max) (-1.190,1.720) (-1.403,1.393) (-1.879,1.385) (-1.168,1.351)Controls:Individual Charac. YES YES YESAverage Charac. YES YESVillage Charac. YESNotes: This table repeats the randomization described in section 2.5.3 one thousand times.The coefficient and thus the standard errors calculated as the mean and standard deviation ofthe one thousand estimates. The dependent variable is income.ated data set, we perform the IV estimation as we did at the subsection 2.5.We repeated this one thousand times and recorded the coefficient from eachrandomization. Finally, we use the mean and standard deviation of thesecoefficients as the coefficient and its standard error of the randomizationtest. Note this method is very similar to the classical bootstrap estimation,but differs in how to generate the repeated observations. The randomizationtest has a good size in the sense that if the second argument is true, therandomized coefficients should be significant as well.The results for male are listed in Table 2.9. It clearly indicates thatwe cannot reject the assumption that the true value of the coefficients arezero under various controls. Thus, Table 2.9 negates the argument that ourresults are possibly driven by the noise in our data set.2.5.4 The Exclusion AssumptionThe validity of the identification strategy that was used in the above sec-tion rests on the assumption that family name composition is a legitimate452.5. IV Estimatesinstrument for social capital in the earnings equation. It becomes clear inthe first stages that the family name composition is related to social capitalbecause of lineage. However, for the validity of the IV strategy, it must betrue that the family composition is uncorrelated with the residual in theincome equation 2.2. In other words, if family name composition affectsincome other than social capital, our approach is called into question. Weargue the exclusion assumption holds for the following reasons:First, as it has been shown in the first stage robustness analysis whereHan vs. non-Han and south vs. non-south Chinese villages were employed,lineage does affect social capital, even though as an institution lineage wasabolished by the Communist party in 1950s and replaced by village partybranches.Second, it is necessary to distinguish between family names and familyname distribution. It would be true that the family name of a villager canaffect her personal earning in some way other than through social capital,but it does not necessary indicate that the distribution of family names issignificantly correlated with income through other channels.Third, it is possible that family name composition could affect earningsthrough effects on the provision of public goods and risk sharing contractsetc.. If family name distribution is correlated with a village?s capacity toprovide public goods or sign risk sharing contracts, the exclusion assump-tion is not warranted. However, its effects on estimates are not clear due tothe fact that the instrumental variables in our analysis are constructed fromthe distribution of family names. Further, if such effects are substantial,we should expect substantial differentials among the estimates from frac-tionalization index and polarization index since these index are nonlinearlycorrelated. However, the IV estimates clearly shows that is not the case.Though risk sharing is difficult to measure, the CGSS 2005 data set doesprovide a good measurement on village level public goods provision, whichincludes expenditure on roads and bridges, agriculture infrastructure, elec-tricity and communication etc.. To test this possibility, Table 2.10 reportsthe OLS estimates of public expenditure on the constructed instruments atvillage level. None of the coefficients of either instruments is significant.462.5. IV EstimatesTable 2.10: Does Family Name Distribution Affects PublicGoods Provision?(1) (2) (3)Panel A: Fractionalization IndexFractionalization Index .34(.57) .14(.59) -.08(.63)Controls:Average Charac. YES YESVillage Charac. YESR2 .001 .033 .013F?value 0.3 1.32 2.97Observations 378 375 344Panel B: Polarization IndexPolarization Index -.41(.63) -.19(.61) -.04(.63)Controls:Average Charac. YES YESVillage Charac. YESR2 .002 .04 .13F?value .43 1.25 3.05Observations 378 375 344Notes: Standard errors are in parentheses. A single asterisk de-notes significance at the 10% level, double for 5%, and triple for1%. The dependent variable is public good expenditure at villagelevel.Also notice the low F values at all regressions. Though this result can notcompletely verify the exclusion assumption, it does show that, at least frompublic good provision perspective, the effect would be very small.2.5.5 Interpretation of the Gender DifferenceOur empirical investigation reveals a striking result: social capital has alarge first order effect on male villagers but no real effect on female villagers.Though a full explanation is difficult, some possible causes can be proposedfrom both the lineage practice and the motivating model.First, women might be less active in village affairs and therefore are lessexposed to the lineage system because they usually are not born in theirhusbands? villages. Indeed, they migrate to the village for the purpose of472.6. Conclusionmarriage. Thus the constructed instruments can not capture the variationof social capital accessible to women since the validity of instruments relieson the claimed connection between family name composition and social cap-ital. In other words, women are less influenced by the lineage system andnaturally their income are less dependent on social capital.Second, the motivating model does predict a significant effect of socialcapital, at least for male villagers. Thus, one way to understand the gen-der differentiation is to trace back to the model and speculate at whichstep we could change the model to produce the result observed. In otherwords, we suspect that the gender difference can be attributed to occu-pational differences, that is, male villagers mainly engage in business thatrequires exchange with a third party, e.g., making a bed as a carpenter inexchange for money, while female villagers work in home production whichdoes not involve a third party, i.e., raising backyard chickens and sellingeggs. Eggs-selling, which can also be done by husbands, is of course anexchange, however raising chickens, which usually is limited to wives, is themore essential part of the chicken business. Formally, male villagers? workis more social and thus reputation based and female villagers? work less so-cial and thus less influenced by social capital. The above interpretation ismerely a conjecture and more detailed work is necessary. Since the presentdata does not include occupational information, we leave this task to futureresearch.2.6 ConclusionThere is now considerable evidence that social capital, defined as trust instrangers or membership of associations, is an important determinant ofeconomic performance and financial developments. For example, Guiso etal. [2004] identified the effect of social capital on financial developmentby exploiting differences in social capital within Italy. They found in hightrust areas, households are more likely to use cheques, have higher access toinstitutional credit and rely less on informal credit, even after controlling forsocial environmental variables. As other work that links social capital and482.6. Conclusioneconomic outcomes, research work investigating the causal effect of truston individual economic achievement usually suffers from the endogeneityproblem.Our identification strategy is to exploit the variation in the family namedistribution of Chinese rural villages. The unique cultural tradition of thelineages offers credible exogenous variation in social capital as measured astrust in strangers at village level. The abolishment of the lineage institutionas a formal institution in the 1950s and our focus on the distribution ofsurnames reinforce the confidence in the exclusion assumption. We constructtwo indices from the distribution to instrument for village level social capital.The first stage analysis displays a significant and robust relationship betweenthe indices and trust. Our GMM estimates mimic the OLS results and findthat a one standard deviation change in trust is almost equivalent to two tofour years of schooling in terms of the causal effect on earnings. However,this effect only exists among male villagers, not among female villagers.Robustness to weak IV estimates confirm the above conclusion.Our conjecture for the gender differentiation in the economic effect oftrust is that male villagers? earning are mostly from free market trade whilefemale villagers? are mostly from home production. We build a simple modelto show that trust determines a villager?s trade volume and thus income.Since home production doesn?t rely on trustworthiness, our model providesa simple explanation for the difference.This paper is a first step to understanding trust and individual earnings.Needless to say, much more empirical and theoretical work is still required.For example, other historical events that might have some impact on socialcapital and income are not accounted in our analysis. These events includecivil war, the Great Famine and the cultural revolution etc.. All of thesecan bias the results if they are correlated with the name distribution. Nodiscussion to explore how the historical contest of lineage rivalry and coop-eration can effect the current trust level. We adopted two indices, namelyfractionalization and polarization, to capture the effects of lineage relationson trust. Are these two indices the most appropriate? If not, how to con-struct other indices to build reasonable behavior assumptions? We believe492.6. Conclusionthese topics are fruitful areas for future research.50Chapter 3Identifying SubjectiveExpectation Bias: Methodand Evidence from theHealth and RetirementStudy3.1 IntroductionEconomic models of choices under uncertainty typically assume people choosean action to maximize expected utility by combining their preferences andsubjective probability distributions over uncertain outcomes. Without datarich enough to include subjective expectations, researchers need to furtherassume rational expectations in order to identify interesting preference pa-rameters and predict choices. This compromise is necessary since the ob-served choice behaviour would be consistent with many alternative com-binations of preference and expectation, which was elegantly discussed byManski [2004]. Between theory and practice, one critical question is whetherpeople form subjective expectations in the way described by the rational ex-pectations assumption.Empirical tests have generated mixed results. In an experimental envi-ronment, many psychologists have reported various subjective expectationbiases. Tversky and Kahneman [1974] found various biases of judgment un-der uncertainty and Tversky and Kahneman [1981] reported that the eval-513.1. Introductionuation of probabilities depends on the way a problem is framed. Hogarth[1975] offered a comprehensive review on cognitive processes and subjectiveprobability assessments and concluded that people are ill-suited for assessingprobability distributions because of limited capacity to process information.Though these researchers reported various abnormal observations on subjec-tive probability evaluations, it is not clear how these abnormalities departfrom rational expectations. In contrast, many economic researchers havefound support for the rational expectations assumption. Bernheim [1988]found that seniors from the Retirement and Health Study responded ratio-nally to new information, though the subjects did not extract all availableinformation to form expectations. Smith, Taylor and Sloan [2001] evaluatedthe relationship between subjective beliefs on mortality and actual deathat the individual level and found remarkable consistency. Ben??tez-Silva andDwyer [2005] tested the retirement expectations of old married Americancouples and concluded they are consistent with the rational expectationshypothesis.This paper proposes a simple definition of expectation bias by exploitingan implication of the rational expectations assumption. The rational ex-pectations assumption states that people should hold an objectively correctexpectation conditional on what information they possess (Manski [2004]).This implies that the same information should be weighted identically inprojecting both objective and subjective probabilities. Accordingly, expec-tation bias is defined as the inequality of the weights.To illustrate, consider the problem of seniors in the Health and Retire-ment Study (HRS) data set. The seniors were asked to assess their individualprobability of entering a nursing home within five years. Consider the in-formation represented by a variable reporting one?s health status. To makethings simple, let us assume health is measured properly. Under the rationalexpectations assumption, the coefficient of health status should be ? in pro-jecting the subjective probability, if the coefficient in projecting objectiveprobability were ?.Of course, the objective probability could never be known outside acontrolled experimental environment. What is observed is the realization of523.1. Introductiona random event. Thus, to assess rationality and identify expectation bias, weneed to substitute the observed realizations for objective probabilities. Thesubstitution brings an identification issue, which is whether the coefficientsare still comparable between the equations of objective probability and ofex post realization, by introducing interim events between the formation ofexpectations and the realization of the random event. This paper shows thatas long as an orthogonality condition holds, identification can be successfullyachieved.Although subjective probability can be elicited through survey questions,the measurement error could be a serious issue for any formal analysis.We find that the self-reported subjective probability of the seniors in theHRS dataset is measured reasonably well, which is consistent with manyprevious evaluations of the same dataset. For example, Hurd and McGarry[1995, 2002] found the HRS respondents? self-reported subjective probabilityof survival aggregates to actual population probability and predicts actualsurvival.We find evidence for expectation biases among both female and maleseniors. The female seniors do not correctly internalize age informationinto their subjective assessment of the probability, while the male seniorsinternalize income information poorly. The estimated coefficient of age 1in projecting subjective probability for the female sample at various surveyyears is about 0.02 while the estimated coefficient in projecting objectiveprobability is about 0.05. The estimated coefficient of income for the malesample in projecting subjective probability is about 0.01 while the corre-sponding coefficient in projecting objective probability is about ?.25. Thehypothesis of identical age coefficients is rejected at less than a 0.1% levelof p-value and a similar hypothesis about the income coefficients is rejectedat less than a 5% level of p-value.Some potential threats to our conclusions are analyzed, which providesprimary evidence against the suspicion that the identified biases could bedriven by wishful thinking of seniors or a violation of the orthogonalityassumption due to public policies regulating nursing home facilities. Discus-1Which is measured as the true age in years minus 65.533.2. Definition and Identification Assumptionsion on the sources of the biases shows that the age bias and income bias canonly slightly be attributed to cognitive factors or a misconception betweenprobability and risk aversion.The remainder of the chapter is structured as follows. We propose thedefinition of expectation bias in section 3.2. To do so, we briefly review somerelated work before discussing the identification assumption. The estimationframework is also briefly presented. Section 3.3 provides a background tothe data and a descriptive analysis. Choosing covariates and evaluatingsubjective probability data are also presented there. Section 3.4 gives themain results of the paper, which is followed by robustness analyses. Section3.5 discusses two potential sources of identified biases. Section 3.6 concludes.3.2 Definition and Identification AssumptionThis section is the major theoretical part of the paper. It first briefly reviewsrelated literature before formally proposing a definition of expectation bias.The definition compares the estimates of the ex ante objective and subjec-tive distribution functions. Since the ex ante objective probability is notobservable in most cases, our identification of bias hinges on a weak or-thogonality assumption, which essentially states that the unexpected partof interim events between ex ante and ex post are orthogonal to populationcharacteristics. Following that is a short discussion in model specificationand equality hypothesis test.3.2.1 A Belief Literature ReviewUnderstanding how people form expectations is both interesting to studentsand important to policy makers. In academia, de Finetti [1969] stated ?thetrue problem consists in the investigations concerning the ways in whichprobabilities are assessed by more or less educated people...? Since forward-looking individuals make decisions based on an assessment of unknown fu-ture states, better understanding its underlying mechanism can improve theefficiency of public policies in the fields of schooling (Manski and Dominitz543.2. Definition and Identification Assumption[1996] and Attanasio and Kaufmann [2009]), saving (Hamermesh [1985] andDominitz and Manski [1997]) and retirement (Chan and Stevens [2004]) etc..As Manski [2004] argued, it is theoretically impossible and practically infe-rior to infer preferences and predict future choices without invoking subjec-tive probability. In experiments, Nyarko and Schotter [2002] and Bellemare,Kro?ger and Van Soest [2008] found models using subjective probability datacan generate better sample predictions than various models where agentsare presumed to form expectations rationally.The work from different fields has produced different conclusions on therationality of expectations: behavioural and psychology researchers over-whelmingly find various biases in subjective probability assessments, whilemost economists find evidence supporting rational expectations. The dis-crepancy arises from different understandings about rationality in assessingsubjective probability. The psychology literature focuses on finding obvi-ous violations of the fundamental principles of reasoning and judging underuncertainty, while the economic mainly focus on how people update theirsubjective assessments upon the arrival of new information. For example,Tversky and Kahneman [1974] described three heuristics that are employedin making judgments under uncertainty. Bernheim [1990] formally exploredhow the elderly form expectations by analyzing the response of self-reportedexpectation date of retirement to new information about Social Securitybenefits using the Retirement and Health Study data.Instead of directly examining the process by which people form expecta-tions, many literatures shift to evaluating subjective probability assessmentsor more broadly subjective judgments by various criteria. This approach fo-cuses on what people really do, and generates insights about what peopleshould do in an ideal environment. Roughly speaking, these literatures canbe classified into four groups. The first mainly compares subjective judg-ments and outcomes in clinical studies. In his seminal book, Meehl [1954]examined 20 studies that compared clinical and statistical predictions andconcluded that the judgment of a statistical model outperforms trained ex-perts. Meehl?s book motivated much other work in the field. The generalconclusion is actuarial judgment generally does better than human judgment553.2. Definition and Identification Assumption(see Grove et al. [2000] for a meta analysis). The second group focuses oncomparing judgments on various fundamental statistics and the true statis-tics. The statistical concepts are usually mean, variance, independence andrandomness. The main conclusion here is that because of cognitive limits,people are ill-equipped for assessing statistics. The third focuses on testingthe rational expectations assumption in relation to updating expectationsbased on Bayes? law. Bernheim [1990] perhaps is the first paper examin-ing the rational expectations assumption using micro data. Ben??tez-Silvaand Dwyer [2005] tested the rationality of retirement expectations followingBernheim?s framework.The last group consists of some recent papers attempting to understandhow people weight various information in forming subjective probability as-sessments. Hurd and McGarry [1995] studied how the elderly assess theprobabilities of survival and found that the subjective probabilities ?covarywith other variables in the same way actual outcomes vary with the vari-ables?. Holden, McBride and Perozek [1997] examined how personal charac-teristics and health conditions influence subjective expectations of nursinghome use and found that both men and women incorporate what is knownabout nursing home risks into their own subjective expectations. Both ofthese works, however, draw conclusions on the rationality of subjective prob-ability assessments based solely on the signs of the coefficients that predictthe subjective probability. For this reason, these works represent an essen-tial departure from previous works by using real survey data and checkingthe sign of a covariate in both the regressions.This paper expands the information weighting idea by exploiting a coreimplication of rational expectations. While these papers compared the signs,in this paper we compare the magnitudes. As Manski [2004]emphasized?the (rationality) assumption per se does not specify the expectations thepersons hold. It asserts only that persons hold objective correct expecta-tion conditional on the information they possess.? Therefore, we impose astrict condition on rational expectations. The condition requires an identicalweight of an covariate in projecting the subjective and objective probabilityunder appropriate specifications and assumptions.563.2. Definition and Identification Assumption3.2.2 DefinitionAssume Y is the interesting random event, e.g., entering a nursing home inthe future. Under the general cases where Y is a continuous random variable,we end up with a very complicated distribution function. To simplify theanalysis, we consider only the simplest case of a dummy variable Y, thatis the realized values of Y can only be zero or one. Thus, the distributionfunction can be concisely expressed as the conditional expectation functionand the conditional probability function: E(Y|X) = Prob(Y|X).Consider the data generating process (DGP) described by an objectivedistribution function G(Y|X,Z : ??), where ?? is the parameter vector de-scribing the relationship between random vectors Y and X,Z. Withoutknowing the objective distribution function, a researcher observing only Xspecifies the objective conditional distribution function F(Y|X : ??), where?? is the parameter vectors characterizing the objective distribution of Yconditional on random vector X. A rational individual, diagrammed below,observing the history and the random variables X and Z is interested inthe data generating process and forms her subjective distribution functionG(Y|X,Z : ?o). Again, without knowing the subjective distribution func-tion, the researcher specifies the subjective conditional distribution functionF(Y|X : ?o), where ?o is the corresponding ?? in the subjective distributionfunction and characterizes the individual?s belief on the distribution of Yconditional on observed random vectors X.HistoryObserve CovariatesX and Z// Expectation YIn this scenario, rational expectations and expectation bias are definedas:Definition. Rational expectations is defined as ?o = ??.With this definition, we proposeLemma. Under rational expectations, ?o = ??.Proof. Under rational expectations, ?o = ??. Thus, G(Y|X,Z : ??) =G(Y,X|Z : ?o). Define M(X|Z : ??) such that G(Y|X,Z : ??) = F(Y|X :573.2. Definition and Identification Assumption??) ? M(X|Z : ??). Similarly for M(X|Z : ?o). Under rational expecta-tion, we also have ?o = ??. Thus, under rational expectations, we have?o = ??.The lemma essentially states that parameters describing the relation-ship between X and Y should be identical regardless of the specification ofF if individuals form rational expectations. Since the true structural func-tion G is not known generally, the lemma offers a tool to evaluate rationalexpectations without imposing strong constraints on F . Thus, we haveDefinition. Subjective expectation bias of Y in terms of X is defined as?o 6= ??.It should be first emphasized that the definition only focuses on the com-parison between the ex ante objective and subjective distribution functions.While the ex ante subjective distribution can be elicited from a survey, theex ante objective distribution is given by the DGP and not known outsideof a controlled experimental environment. Nevertheless, the definition cap-tures the idea that explanatory variables should have the same weight inprojecting subjective probability as in the objective one.The definition implies that conditional subjective distribution of Y onX and Z, Prob(Y = 1|X,Z), is identical with the true distribution func-tion. Thus, the rationality of expectations requires optimal perception,which of course can be a result of evolution as many have argued. Note,however, the optimal perception is also the correct perception, where cor-rectness indicates identical with the true distribution of the DGP. If per-ceptions were not correctly chosen, there would exist unexploited utilityor profit-generating possibilities within a system. Furthermore, it also im-plies the conditional subjective probability function on the unobservable Z,Prob(Z|X), is identical with the objective one. Since Prob(Y = 1|X) =Prob(Y = 1|X,Z)?Prob(Z|X), rational expectations leads to the conclusion?o = ??.The idea that a covariate used to form subjective expectations should beutilized the same way as it predicts the objective distribution of Y is a very583.2. Definition and Identification Assumptionnatural expansion and development of the conventional rational expectationsassumption, which requires the coincidence of subjective probability andobjective probability conditional on information. Since objective probabilityis mostly unobservable, the canonical rational expectations assumption isprincipally limited as a solution concept in empirical work. By contrast,the proposed definition does not directly compare subjective and objectiveprobability and thus can be potentially adopted in many research fields. Asa compromise, however, the definition does require the presence of covariatesX.The idea is not novel in social sciences. McClland and Bolger [1994] andHurd and MaGarry [1995] analyzed how aged people form their longevityexpectations by using the HRS data set and noticed that ?most remarkable,however, is that they (subjective expectations) covary with other variablesin the same way actual outcomes vary with the variables.? The proposal ofattribute substitution in judgment first offered by Tversky and Kahneman[1974] and then revised by Kahneman and Frederick [2002] shares almostthe same opinion about how people should make judgment under uncer-tainty, though it highlights the abnormal phenomena of representativenessand availability biases. In a nut shell, attribute substitution occurs whenpeople, in making judgment, assess a specified attribute by substituting itwith another attribute, which comes more readily to mind, e.g., heuristicsand anchors. The identification of attribute substitution overwhelminglydepends on being capable of finding a substitution attribute, which canbe elicited in experimental conditions but is difficult otherwise. In part,this makes many people suspect that the reported judgment bias is onlya result of the way the statistical information was presented to respon-dents(Gigerenzer [1991] and Hoffrage et al. [2000]).Our definition exploits the implication of the rational expectations as-sumption that people should use information correctly and accurately: theyshould form subjective probability the same way as the covariates predictsobjective probability. The real DGP is not known to us, nor is its struc-tural form. Nevertheless, the parameters characterizing the reduced formof objective and subjective distribution functions should be identical if the593.2. Definition and Identification Assumptionrational expectations assumption holds.Bias is identified as the discrepancy between ?? and ?o. The identifi-cation of ?o is guaranteed if we know the subjective probability of Y andcovariates X, with the aid of the specification E(Y|X : ?o). The subjectiveprobability cannot be directly observed, but can be elicited. Since the 1990s,various surveys have asked interviewees about expectations, e.g., the Healthand Retirement Study, which this paper focuses on, and the 1997 cohortof National Longitudinal Survey of Youth. Similarly, the identification of?? can be achieved once we know the conditional probability, which can beestimated from the realized values of Y. Notice, however, even with theidentification of these two parameters, identifying biases still need to assurethat the parameters are comparable.Last, though bias is stated as ?in term of X?, the interpretation can bemuch more flexible in practice. More concretely, consider two predictorsx1 and x2. Assume ?o1 = ??1 and ?o2 6= ??2 , therefore the source of theexpectations bias is incorrectly internalizing information from x2. Furtherassume x1 is observable and x2 is not observable in the data set and thetwo predictors are correlated x1 = ?x2 + ?. Under regular conditions, theprobability limit of the estimate of ?o1 and ??1 would be ?o1 + ??o2 and ??1 +???2 . Since ?o2 6= ??2 , the estimates of the coefficient x1 in the objectiveand subjective probability function will not be equal. Without knowing x2,we thus conclude there is expectation bias in terms of x1. In this sense,identifying a bias in terms of any covariates can actually point to the samebias sources in some extreme cases 2. According to this argument, we shouldnot overly emphasize a specific predicator in explaining bias. Of course, onewould suggest testing the equivalence of all the coefficients instead of onlytesting the coefficient of a specific covariate. We do not because of practicalconsiderations. First, it is convenient to frame the discussion in terms ofsome specific covariates. Except in extreme cases, identifying and analyzing2One example of an extreme case is the real expectation bias only exists for the un-observable covariate z and all other observable covariates xs are correlated with z. Then,depending on the specification of probability distribution equation, we would identify biasin terms of various x.603.2. Definition and Identification Assumptionbias in term of a specific covariate could help to shed light in exploring theroot causes of the bias. Second, testing of the equivalence of all coefficientsdemands stronger assumptions on the distribution of estimates.3.2.3 Identification AssumptionThe identification of expectation bias needs to compare ?o and ??, whichrequires observing both the ex ante subjective and objective probabilities.Since the objective probability is not observable in most cases, ?? has to beinferred from the ex post realization of Y, Y .Replacing the ex ante objective probability with the ex post realizationintroduces interim random events that would affect the realization of Y. Thefollowing diagram shows how realization proceeds, following the formationof expectations. As above, an individual forms expectations after observingoutcomes and covariates X and Z, where X is observable to researchers andthe individual and Z is only known by the latter. Before the realization ofY, Y , some observable random event W and unobservables  can affect therealization of Y but not the ex ante objective probability Y. Since we arenot directly interested in W and , we group them as random variable ? 3.HistoryObserve CovariatesX and Z// Expectation YRandomness? = (W, )// Y (Realization of Y)The above concern is central to the identification strategy. In the fol-lowing analysis, we first propose a strong assumption and then a weak as-sumption for identification. With the introduction of interim random events,the identification becomes under what conditions the ?? can be consistentlyestimated by observing the realizations Y . Notice ?? characterizes the re-lationship between Y and X in F(Y|X : ??). After introducing the interimevents ?, consistency requires ??be identical to ??, where ??(?) charac-terizes the relationship between Y and X (?) in F(Y |X, ? : ??, ?). This3Of course, we could control W in the objective regression if it is observable. But itwill not eliminate the consistency problem since  is never observable.613.2. Definition and Identification Assumptionidentification requirement can be better understood in two steps. First, itrequires the value of ?? be preserved from F(Y|X : ??) to F(Y |X : ??),which is always true. Second, it further demands the equivalence of ? fromF(Y |X : ??) to F(Y |X, ? : ??, ?). Depending on the specification of F(),different assumptions will be needed to preserve the value of ??. To simplyour analysis, we focus on linear models.Assumption. Orthogonality to interim information, i.e., X is orthogonalto ? or XT ? = 0.Orthogonality to interim information is a weaker statement than exo-geneity. It allows endogeneity to the the extent that interim informationis orthogonal to X, the population characteristics. To understand it, let usconsider the example where we are interested in the probability of an indi-vidual aged 40 entering in a nursing home facility at age 80. Between age40 and 80, many health and social factors will affect the realization of theinterested event. The orthogonality to interim information requires eventshappening between age 40 and 80 are unrelated with population character-istics at age 40. This is a strong assumption and is a sufficient condition foridentification. We later relax it to a weak condition. With this assumption,we have in the linear caseClaim. If the orthogonal to interim information assumption holds, ??= ??,where ??describes the relationship between X and the realization Y and ??describes the relationship between X and objective probability Y.The proof can be achieved as follows. Assume the model without anyinterim events is Y = X?? + ? and the model with the interim events isY = X??+ ?? + ?. Obviously ?? = (XTX)?1XTY=??+ (XTX)?1XT? +(XTX)?1XT?. So, ?? = ??as long as the orthogonal to interim informationcondition holds.Now, we discuss the weak condition for identification. The orthogonalto interim information assumption is strong because it ignores the fact thatindividuals form expectations on interim events ? while forming expectationson Y. If so, the information included in ? but correlated with X, E(?|X),623.2. Definition and Identification Assumptionshould have no impact on identification. Hence, the orthogonal assumptionreduces toAssumption. Orthogonality to residual information, i.e., X is orthogonalto ? ? E(?|X), where E() is ex ante subjective expectation.Similarly, we haveLemma. If the orthogonal to residual information assumption holds, ?o =??under rational expectation, where ?o describes the relationship between Xand subjective probability Y and ??describes the relationship between X andthe realization Y .The proof of the lemma parallels that of the above claim. Let themodel of subjective expectation be Y = X?? + ? and the objective ex-pectation model with the interim events be Y = X??+ ?? + ?. Underrational expectation, let ? = X? + ?, therefore Y = X(?? + ??) + ? andY = X(??+ ??) + ? ? E(?|X)? + ?. Following the same argument in theabove discussion, we have ?? = ??if X is orthogonal to the residual infor-mation ? ? E(?|X).Unlike its strong pair, the orthogonal to residual information assump-tion only requires being uncorrelated with unexpected information. In theabove example, the expected part of the interim events will have no impacton identification. What matters are the unexpected part of the events fromage 40 to age 80, e.g., health shocks arriving after age 40. If the shocks areorthogonal to population characteristics at age 40, it is still possible to suc-cessfully identify bias. Cases where the weak condition is violated do exist.Consider the negative impact of smoking, which was largely unknown bysociety in the 1960s. Assuming smoking behaviour is correlated with otherindividual characteristics, which seems acceptable, the weak assumption isviolated if smoking happens at the interim period. The next paragraphoffers more detailed discussion on threats to the assumption.Threats to the Weak Assumption The assumption is crucial for iden-tification. Should it be violated, it may negatively affect any conclusions on633.2. Definition and Identification Assumptionexpectation bias. Thus, it is valuable to know what factors can potentiallyinvalidate it.Most of the new events X? of X would not be a threat to the weak orthog-onality assumption. Usually, X? is generated conditional on the outcome ofX. The correlation will violate the strong assumption, but not necessary theweak one. Following the previous example, health status between age 40and 80 can be correlated with health status at age 40, therefore the strongorthogonality assumption is violated. However, the weak orthogonality stillholds if the unexpected shocks to health status between age 40 and 80 arenot correlated with health status at age 40.A potential threat to the orthogonality between X and ? could be moralhazard. However, in the nursing home market, it is hard to justify sinceentering into a nursing home facility usually needs at least two certifiedADLs from professionals. Further, a threat to the strong assumption doesnot threaten the weak assumption, which requires moral hazard behaviourto be unexpected. This seems even harder to justify. The major sourceof the correlation between X and ? ? E(?|X) is unexpected public policies.Typically, to achieve some objectives, an interim policy is designed basedon the statistics of covariate X across observations. If this policy is notexpected by individuals and therefore not integrated into expectations, theunexpected policy design could introduce a correlation between the policyand X, violating the weak assumption. In the background of long termcare, if an unexpected interim public policy that is based on the attributesof seniors in our sample is implemented, it might be a concern from anidentification perspective.3.2.4 Model and Test MethodIn linear models, the conditional probability function of F can be writtenas:Prob(Y = 1|X) = G(X?o) (3.1)andProb(Y = 1|X) = G(X??) (3.2)643.2. Definition and Identification AssumptionGiven the above functional forms, it is tempting to estimate ?s simul-taneously by using a bivariate probit model. However, the bivariate normaldistribution of the error terms 4 is inappropriate as the range of subjectiveprobability Prob(Y = 1|X) is the interval from 0 to 1, not the points 0 and1 as in the bivariate Probit model case.Estimating ?s simultaneously is relatively efficient. This can actually beachieved by applying a seemingly unrelated estimation (SUR) method withthe linear probability assumption of G(X?) = X?. As with all linear prob-ability models, due to the assumed constant marginal effects of explainingvariables, estimators can easily fall outside the unit interval.The second choice is to estimate equation 3.1 and 3.2 separately. Aprobit model can be applied to estimate ?o. To estimate ??, we adopt thefractional probit method proposed by Papke and Woodridge[1996]. Thefractional probit method was originally proposed to deal with fractionaldependent variables which range from 0 to 1 as well.Comparing with other procedures like Berkon?s minimum chi-squaremethod advocated by Amemiya [1981], the fractional probit method is veryattractive because it doesn?t require any ad hoc transformations of the ex-treme values of 0 and 1. Recovering the regression function to get theprobability can be easily handled as in a probit model. More precisely, theprocedure maximize the following log-likelihood functionl(?) = Prob(Y = 1) ? log[G(X??] + (1? Prob(Y = 1)) ? log[1?G(X??)]which is identical with the log-likelihood function of a probit model, exceptthe range differs. Under some regular conditions, consistency and asymp-totic normality can be easily derived.To test the equivalence of the coefficients from the probit and fractionalprobit estimation, a similar idea to Hausman [1978] is applied. In the Haus-man test, where one estimator is efficient and one is consistent, the co-variance of the two coefficients asymptotically equals to the variance of the4Actually, the bivariate probit model requires the potential error terms be bivariatenormal, not the error terms we presented in equation 3.1 and 3.2.653.3. Background, Data and Description Analysisefficient estimator. Without the efficiency hypothesis, we calculate the co-variance of the coefficients, using the sandwich form proposed by White[1996]5. We only report the results from the probit and fractional probitmethod. The linear SUR method generates similar results and thus is omit-ted.3.3 Background, Data and Description Analysis3.3.1 HRS and Expectation QuestionThe Health and Retirement Study (HRS) is a biennial, longitudinal surveyof Americans, first conducted in 1992. It was originally designed to analyzeretirement transition and thus initially only included the cohorts born be-tween 1931 and 1941. In 1998, the HRS was merged with another survey,AHEAD, of individuals born in 1923 or before to cover all Americans over50. In the following years, other sub samples were successively added tomake it representative of this population. As one of the largest and mostambitious social science projects undertaken in the USA, the HRS is care-fully designed to reflect both academic and policy interests in the area ofaging and retirement planning. The comprehensiveness of the survey con-tents makes it suitable to answer a variety of different questions. Its coresections include demographics, physical health, household mobility, fam-ily structure, work history, disability, income, wealth, insurance, cognitiveability, expectation, and retirement planetc.. Other experimental modules,including risk preference and parental wealth among others, are randomlydistributed to a fraction of the respondents.One unique feature of the survey makes it well suited for the purposeof studying subjective expectation bias. It asks the respondents about sub-jective expectations of future events, including longevity, living standardafter retirement, working after retirement, leaving a bequest, and movingto a nursing home in five years etc.. Though questions about expectationshave been asked in other earlier surveys, e.g., the Retirement History Study,5For a detailed discussion on the test in Stata, see Weeise[1999].663.3. Background, Data and Description Analysisthe absence of carefully designed instruments makes the data from thosequestions limited in empirical work. Bernheim[1989, 1990] found the inter-pretation of expectations to be somewhat problematic: do people think ofthe mean, the mode, or the median when they are asked about their ?ex-pectation?? Instead of vaguely asking the respondents ?expectation?, theHRS is carefully designed to ask the respondents? subjective estimate of theprobability of some categorical random variable being true, e.g., the chanceof living to 85 or the chance of working full-time after retirement. Anotheradvantage of the expectation of a categorical random variable is that an-swering it may require less cognitive effort, which might be crucial sincemost surveys present no incentives for answering questions. Furthermore,the HRS strictly restricts the answers in numerical form ranging from 0 to100 percent, which logically gives it a probability interpretation. Since dis-putes about subjective data easily arise, we thus describe the design of theexpectation questions in detail.To familiarize respondents with the basics of probability, the expectationsection always begins with the following introduction:Next we would like to ask your opinion about how likely youthink various events might be. When I ask a question I?d likefor you to give me a number from 0 to 100, where ?0? means thatyou think there is absolutely no chance, and ?100? means thatyou think the event is absolutely sure to happen. For example,no one can ever be sure about tomorrow?s weather, but if youthink that rain is very unlikely tomorrow, you might say thatthere is a 10 percent chance of rain. If you think there is a verygood chance that it will rain tomorrow, you might say that thereis an 80 percent chance of rain.Before continuing the survey, the respondents were given the opportunity todo a warm-up exercise:Let?s try an example together and start with the weather.What do you think are the chances that it will rain or snowtomorrow?673.3. Background, Data and Description AnalysisTo help the respondents better answer the questions, all questions areimmediately followed with a graphic of an interval scaling from 00 to 100.It is marked with words ?Absolutely No Chance? at 00 and ?AbsolutelyCertain? at 100.After completing the expectations questions, e.g., chances of leaving anyinheritance and of working for pay for sometime in the future, the respon-dents aged over 65 after the survey year 1998 were asked the question:What is the chance that you will move to a nursing home inthe next five years?which is immediately followed with a definition of a nursing home. Therespondents? answer would reflect their assessment of their health status,financial status, insurance coverage, and other related information.Besides the subjective probability, another dependent variable is thedummy variable of the realization of the interesting event. Because of thelongitudinal nature of the survey, we can observe if the state of interest wasrealized, i.e., they moved to a nursing home in five years.Comparing the frequency of the realized events and the self-reportedprobability would be interesting, but we must first consider how to selectthe covariates x. There are two principles that we should be mindful of.First, the covariates should be obvious and known by the respondents. Thisis a natural requirement since rationality is conditional on the informationthe subject possesses. Second, the covariates should be predictive of theobjective probabilities. The better the predictive power, the more accuratethe estimates, which will be crucial later for testing the null equivalencehypothesis.3.3.2 Choosing CovariatesThe identification of expectation bias relies on comparing the values of co-variates X in predicting expectations and realized events. Many factors canaffect nursing home utilization. Demand side factors include the health sta-tus, household income, family structure, race and age of the individual. The683.3. Background, Data and Description Analysissupply side factors are the cost of nursing home care and local regulatorypolicies etc.. However, including supply side factors in our analysis is dif-ficult because of the lack of sufficient variation. The RAND version of theHRS does not include data on state of residence. If it were available wemight be able to integrate more supply side policy factors such as those thatvary at the state level.Health status can be measured in different ways. One is the subjectiveevaluation of the respondent. However, measurement error could contam-inate the analysis. Thus, we focus on objective measures of health status.The first indicator is the health condition, which is the sum of the followingfactors: high blood, pressure, diabetes, cancer, lung disease, heart problem,stroke, psychological problem, and arthritis. The other health indicators arethe activities of daily living (ADLs) and the instrumental activities of dailyliving (IADLs). ADL functions include bathing, walking, dressing,eating,and getting in/out of bed. IADL functions range from using a map, a cal-culator, and a telephone, managing money, taking medication, shopping forgroceries, and cooking meals. Both ADLs and IADLs are good predictors oflater long term care utilization and are widely used in policy underwritingand pricing. Some researches suspect that many elderly people are not fullyinformed of the prediction power of ADLs. Thus, testing the expectationbias in terms of ADLs and IADLs can shed some light on the debate.Another health indicator that is highly related with nursing home utiliza-tion is the respondent?s past nursing home stay experience. Past experiencecould be a good predictor about future use of nursing home facilities, butcan individuals properly use this information?Household income can influence nursing home care utilization becausethe costs are high and are not covered by general insurance policies, e.g.,Medicare or health insurance sponsored by employers. In practice, mostseniors depend on Medicaid to pay for their nursing home bill. In orderto establish Medicaid eligibility, the seniors must meet both the householdincome and non-housing asset requirements. For seniors with high income,Medicaid requires the seniors spend down their income below some level toqualify Medicaid. After that, Medicaid only pays the remaining bill if the693.3. Background, Data and Description Analysisspending down income is not enough to cover the cost. In this situation,if seniors anticipated a need for nursing home care, they would give assetsabove Medicaid limit to children, increasing current consumption. Thus,the measurement of assets may not be accurate because of manipulation.Thus, assets are excluded from the analysis. The results are not sensitiveto the inclusion of various asset measurements, though. Years of schoolingand race are also adopted as other measures of income and social status thataffect nursing home utilization.Family structure could affect nursing home care utilization either throughits effects on reducing body functional disability or by providing a substituteto nursing home care. We use number of children as a measure of familystructure. Another potential substitute to formal nursing home facility iscohabiting with a partner. In practice, many seniors, specially males, arecared for by their spouse instead of entering a nursing home.Age is correlated with many unobservable health and mental indicatorsand is used for insurance policy underwriting. It is also the most accuratelymeasured and naturally, respondents are fully aware of their age. It wouldbe interesting to examine if the respondents can correctly internalize the ageinformation into the expectation formation process.Unlike other status variables, holding of long term care (LTC) insurancepolicies suffers from self selection. According to the standard theory ofadverse selection in insurance market, those holding an insurance policy arehigh risk individuals. In a recent paper by Finkelstein and McGarry [2006],the authors demonstrate evidence of multidimensional private information inthe long term care insurance market. With regard to subjective probability,we can check if those with the insurance policy have a higher subjectiveprobability or a higher objective probability of entering a nursing homefacility. It would give some useful hints for understanding the insurancemarket.In sum, we have three health status measurements: health condition,ADLs and IADLs; a proxy health indicator of past nursing home stay expe-rience, two family structure measurements, insurance policy holding indica-tor, income, age and race.703.3. Background, Data and Description Analysis3.3.3 Descriptive StatisticsThis section compares the objective and subjective probabilities over vari-ous factors discussed above after presenting the descriptive statistics on thevariables of interest. When it does not give rise to confusion, the word ?prob-ability? is usually used as subjective probability and the word ?chance? isused as objective probability.Table 3.1 provides some descriptive statistics for the wave 4, wave 5and wave 6 samples respectively. Wave 4 is the first year the subjectiveprobability questions were asked in the way described above. The newestavailable wave is wave 9, which means the earliest wave that we have actualnursing home use data is wave 6. In the table, the objective probabilityis measured as the realization of the random event of interest, which takesthe value 1 if a respondent ever entered a nursing home during the fiveyears period and takes the value 0 otherwise. Since not all respondents arerequired to answer the subjective probability questions, we limit our sampleto these who did answer it.Columns 1 ? 3 report mean values and standard deviations for wave6, Columns 4 ? 6 show the descriptive statistics for wave 5 and columns7 ? 9 for wave 4. At each wave, we show the statistics for the whole sam-ple, then males and females sequentially. The first two rows in the Tablereport the average objective probability and subjective probability. In allthe columns, the average subjective probability is very close to the aver-age objective probability, though the former is about 1% higher, which isnot statistically significant. Females are usually better judges of probabil-ity. For males, the subjective probability is always statistically significantlyhigher than the objective probability. The overestimation is about 2%, andsince the overall probability for a male senior is about 10%, the differenceis about 20% in relative value. For females, the subjective probability isalmost identical to the objective one. These statistics clearly show that theself-reported subjective probability actually mimics reality quite well. How-ever, the closeness does not necessarily imply rationality: it only asserts thatpeople hold objective correct expectations conditional on what information713.3. Background, Data and Description AnalysisTable 3.1: Descriptive Statistics(Wave 6) (Wave 5) (Wave 4)All Male Female All Male Female All Male FemaleObjective Probability 0.130 0.100 0.151 0.127 0.093 0.151 0.116 0.088 0.136(0.004) (0.005) (0.005) (0.004) (0.005) (0.007) (0.006) (0.006) (0.007)Subjective Probability 0.139 0.129 0.147 0.142 0.127 0.152 0.124 0.116 0.130(0.003) (0.004) (0.005) (0.003) (0.004) (0.004) (0.003) (0.004) (0.004)Age-65 9.369 8.936 9.675 9.082 8.698 9.351 8.825 8.397 9.130(0.108) (0.145) (0.123) (0.102) (0.137) (0.106) (0.094) (0.120) (0.107)Income 0.433 0.540 0.358 0.410 0.501 0.346 0.375 0.461 0.313(0.011) (0.017) (0.009) (0.009) (0.015) (0.007) (0.008) (0.013) (0.007)Years of Schooling 12.408 12.676 12.220 12.227 12.427 12.086 12.069 12.268 11.928(0.073) (0.102) (0.067) (0.079) (0.113) (0.070) (0.081) (0.100) (0.081)Health Condition 2.087 2.050 2.114 1.949 1.904 1.981 1.844 1.835 1.851(0.018) (0.029) (0.023) (0.019) (0.025) (0.025) (0.021) (0.027) (0.026)ADLs 0.315 0.230 0.376 0.332 0.239 0.397 0.356 0.279 0.411(0.011) (0.013) (0.015) (0.014) (0.014) (0.018) (0.010) (0.016) (0.015)IADLs 0.306 0.215 0.371 0.285 0.188 0.353 0.313 0.234 0.370(0.011) (0.013) (0.013) (0.012) (0.012) (0.015) (0.011) (0.013) (0.014)Nursing Home Expe. 0.027 0.022 0.030 0.019 0.016 0.022 0.018 0.014 0.021(0.003) (0.003) (0.003) (0.002) (0.002) (0.002) (0.002) (0.003) (0.003)LTC Insurance 0.139 0.139 0.139 0.123 0.124 0.122 0.113 0.118 0.110(0.006) (0.007) (0.007) (0.006) (0.006) (0.007) (0.005) (0.006) (0.005)Cohabitation 0.579 0.766 0.447 0.574 0.756 0.447 0.581 0.769 0.447(0.007) (0.008) (0.009) (0.008) (0.010) (0.008) (0.007) (0.009) (0.009)Children 3.199 3.302 3.127 3.164 3.259 3.097 3.110 3.225 3.029(0.041) (0.045) (0.047) (0.042) (0.042) (0.050) (0.047) (0.050) (0.055)Black 0.122 0.122 0.122 0.121 0.114 0.127 0.119 0.113 0.124(0.009) (0.011) (0.008) (0.009) (0.010) (0.010) (0.010) (0.010) (0.011)N 8115 3401 4711 7954 3334 4619 7980 3392 4584Notes: The objective probability is the measured as the realization of the random event. The data source isHealth and Retirement Study, Version 1, RAND Company.723.3. Background, Data and Description Analysisthey possess. Without invoking the relationship between the probabilitiesand the information, it is impossible to evaluate the rationality assumption.The above description might also give the impression that expectationbias exists only among males, not female seniors. This is not the whole storyhowever. Figure 3.1 plots the average subjective and objective probabilityfor different samples over the variable of past nursing home stay. The bluecolumn corresponds to objecitve probability and the red column correspondsto subjecitve probability. The first panel corresponds to wave 6, the secondto wave 5, and the third panel to wave 4. The figure shows that the chanceof entering a nursing home is doubled if a male senior has previously usedone, and the chance is tripled for a female senior. For males, the subjec-tive probability follows the objective pattern, though with some noise. Thestriking feature is for the females, where they generally underestimate theirchance of returning.Figure 3.2 presents a similar plot over holding of long term care (LTC)insurance. Other research has indicated that those who choose to purchaseinsurance tend to be more at risk. This is not true in our figure, wherethese with a long term care insurance policy generally have the same chanceto enter a nursing home as those without. What?s more interesting in ourfigure is the pattern of the subjective probability: it is much higher thanthe true chance for male seniors with an insurance policy. It looks like themale seniors buy insurance policies because they think they are high riskpeople, not because they actually are. However, Finkelstein and McGarry[2005] showed that those with an insurance policy are more risk averse.The similar chart over the age group is given in figure 3.3. Since ageand gender are fundamental social characteristics that influence all kinds ofdecisions and judgments, it is natural to expect the subjective probabilitycould better match the objective for a specific age group. Figure 3.3 doesnot support this idea. Actually, it indicates that the subjective probabilityis much higher than the objective chance for younger seniors. For femalerespondents aged over 80, the subjective probability is far below the truechance they will enter a nursing home. Interestingly, the female sampleseems to support the idea that the self-reported subjective probability has733.3. Background, Data and Description AnalysisFigure 3.1: Comparing Objective and Subjective Probabilities: by NursingHome Experiencesome tendency to regress toward the mean: the subjective probability doesincrease with age, but less than the true chance.Of course, the above figures only present suggestive hints about expecta-tion bias and the rationality of expectations. This evidence is not conclusivesince many other control variables are omitted in the figures. Specially, if thecovariate and unobservables are correlated, any conclusion solely from thegraphic analysis would be misleading without further evidence from formalregression analysis. Though the above statistics also indicate a satisfactoryassessment of subjective probability, a formal evaluation on it can help shedlight on some recent debates on the validity of the subjective measurements.3.3.4 Assessing Subjective ProbabilityPeople are skeptical about subjective survey data, arguably with good rea-sons. Bertrand and Mullainathan [2001] summarized experimental evidenceon how cognitive factors can affect how people answer survey questions. For743.3. Background, Data and Description AnalysisFigure 3.2: Comparing Objective and Subjective Probabilities: by LongTerm Care Insurance HoldingFigure 3.3: Comparing Objective and Subjective Probabilities: by Age753.3. Background, Data and Description Analysisexample, the answers can be manipulated by changing the order of the ques-tions, modifying wording, or adjusting scale. Low correlations of the answersfrom two surveys spaced apart raise the doubt if the answers do measuresomething. Besides cognitive considerations, the objections against subjec-tive data can be classified into two groups: doubts about if people are willingto truthfully reveal their subjective perceptions and doubts about if peoplehave the ability to do so. Unwillingness to truthfully answer subjective ques-tions can be partially a result of the absence of incentives and partially of thelack of verification. Though the absence of incentives is a common featureof all survey data, cognitive cost of answering makes subjective data par-ticularly vulnerable. The extra cost arises because transforming beliefs andperception into numerical values is analytically challenging. As Tourangeau[1984] discussed, answering a survey question is a cognitive process involvingcomprehension of the question, retrieval and encoding of information frommemory, assessment of the correspondence between the retrieved informa-tion and the requested information and finally communication. All theseactivities involve mental and cognitive cost. That there is no way to verifythe subjective answers could lead to moral hazard in reporting the data, i.e.,reporting an easy but unrelated number in order to avoid the cognitive cost.The second doubts if people have the ability to do so. Mental cost, andthe lack of cognitive ability evaluating the uncertainty in numerical formcan impede validity. People would conjure up some figures if they could notexpress the perceived probability in numbers while are forced to do so.In this section, we discuss some of these related arguments against self-reported subjective probability. Formally refuting all the assertions againstsubjective data is impossible. Instead, we focus on the objections mostlyrelated to our data by examining the unwillingness and ability assertion. Weachieve the goal by discussing some recent research supporting the applica-tion of subjective probability data. Following it is an assessment on howself-reported subjective probability can help to predict real nursing homeutilization.Evidence of good performance of using self-reported subjective proba-bility in fitting data have been found by various researchers. Juster [1966]763.3. Background, Data and Description AnalysisTable 3.2: Can Subjective Probability Predict Behaviour?Wave 6 Wave 5 Wave 4Subj. Prob. Age ADLs Subj. Prob. Age ADLs Subj. Prob. Age ADLsCoefficient 0.13*** 0.01*** 0.07*** 0.08*** 0.01*** 0.05*** 0.05*** 0.01*** 0.03***(0.01) (0.00) (0.00) (0.02) (0.00) (0.00) (0.02) (0.00) (0.00)R2 0.01 0.04 0.04 0.01 0.01 0.01 0.01 0.01 0.01Obs. 8295 8295 8295 6244 6244 6244 4892 4892 4892Notes: The dependent variable is the dummy indicating the realization of the random event andthe estimates are from probit estimation. Standard errors in parenthesis. A single asterisk denotessignificance at the 10% level, double for 5%, and triple for 1%.found self-reported purchasing probability outperforms the verbal expres-sion of buying intentions in explaining the automobile purchasing behaviour.Hurd and McGarry [2002] used the same data as our paper to examine howreliable the subjective survival probability is. They found the respondentsactually modify their survival probability in response to new informationand subjective survival probabilities do predict actual survival: those whosurvived in the survey panel reported about a 50% greater probability thanthose who died. As mentioned, Nyarko and Schotter [2002] and Bellemare,Kro?ger and Van Soest [2008] both reported better predictions using sub-jective probability data than merely assuming rational expectations in anexperimental environment. In another paper exploring if seniors can under-stand the risk of moving to a nursing home, Taylor et al. [2005] found thatthe seniors reporting a higher probability of moving a nursing home within5 years were indeed more likely to do so.To assess how well the self-reported probability can predict true risks,we compare the predictive power of the subjective probability with ADLsand age, the covariates popularly applied in industry insurance underwrit-ing. Table 3.2 lists the estimated coefficients in OLS regressions where therealization of entering nursing home is the dependent variable without con-trolling for other variables. Just as was found for the covariates age andADLs status, the coefficients of subjective probability are always statisti-cally significant across waves.773.4. Age Bias and Income Bias3.4 Age Bias and Income BiasThis section first presents the main results, then discusses some threats tothe assumptions underlying the main results. In next section 3.5, we will alsodiscuss the potential sources of expectation bias, which is different than therobustness analysis. The discussion on the robustness answers the concernwhether the identified bias could be a result of a violation of the identificationassumption or an inconsistent estimation of ??, while the sources discussionseeks to find out what factors drive expectation bias.3.4.1 Main ResultsTables 3.3 and 3.4 report regression results of equations 3.1 and 3.2 and theresults of the Wald test for equivalence of the coefficients for various inter-esting covariates discussed above. All of the coefficients listed in the tablesare the original estimates, rather than the marginal effects usually reported,as the magnitude of the coefficients is our focus. Wave 4 to Wave 6 resultsare separately reported, though mixing all waves generates similar conclu-sions. However, separating different waves data provides a cross validationand increases confidence in conclusions.Table 3.3 presents the results from the male sample. Column 1 showsthe estimates from a probit model where the dummy variable of realizationis the dependent variable using wave 6 data, while column 4 uses wave 5and column 7 uses wave 4. Standard errors are listed in parentheses. First,notice in most cases, we find the expected quantitative effect. The senior?sage increases the probability of entering a nursing home, while education hasno effect after controlling for income. Total income significantly decreasesthe chance of entering a nursing home. This might reflect that the richchoose private in-home care rather than a nursing home facility. ADLs havesignificant effect as well, but not IADLs and health conditions. This reflectsthe correlation between ADLs and IADLs and that ADLs measurementsare better predictors of nursing home utilization. A health condition doesnot substantially affect the probability since it is a measure of autoimmunediseases that should be treated in a hospital. Past nursing home stay and783.4. Age Bias and Income BiasTable 3.3: Main Results: Male SampleWave 6 Wave 5 Wave 4Objective Subjective Test Objective Subjective Test Objective Subjective TestAge 0.033*** 0.021*** 5.145 0.030*** 0.026*** 2.693 0.0296*** 0.030*** 0.033(0.000) (0.000) (0.056) (0.010) (0.002) (0.113) (0.006) (0.002) (0.854)Income -0.253* -0.014 5.835* -0.188* 0.023 5.978* -0.26* 0.023 5.185*(0.093) (0.024) (0.023) (0.084) (0.034) (0.027) (0.123) (0.055) (0.032)Years of Schooling -0.012 0.025*** 4.01* 0.015 0.016* 0.336 0.021 0.016 0.095(0.014) (0.003) (0.051) (0.017) (0.010) (0.577) (0.025) (0.016) (0.766)Health Condition 0.06 0.04* 0.71 0.09** 0.07*** 0.44 0.06 0.05* 0.09(0.03) (0.02) (0.40) (0.03) (0.02) (0.51) (0.04) (0.02) (0.76)ADLs 0.141** 0.073** 2.323 0.234*** 0.095* 4.312* 0.194* 0.002 2.484(0.049) (0.026) (0.134) (0.063) (0.044) (0.042) (0.094) (0.056) (0.121)IADLs 0.001 0.010 0.002 -0.015 0.046 0.195 -0.015 0.065 0.262(0.057) (0.034) (0.956) (0.086) (0.053) (0.665) (0.085) (0.062) (0.617)Nursing Home Expe. 0.191 0.164 0.047 0.205 0.154 0.023 0.864* 0.265 2.138(0.172) (0.113) (0.857) (0.254) (0.159) (0.880) (0.374) (0.223) (0.155)LTC Insurance 0.082 0.183*** 1.099 0.10 6 0.232*** 0.758 -0.123 0.255** 3.888*(0.082) (0.043) (0.306) (0.125) (0.063) (0.396) (0.176) (0.082) (0.054)Cohabitation -0.151* -0.024 3.065 -0.068 -0.051 0.015 -0.158 0.033 2.453(0.072) (0.044) (0.092) (0.097) (0.065) (0.923) (0.117) (0.065) (0.132)Children -0.023 -0.042*** 1.473 -0.013 -0.005 0.041 -0.003 -0.013 0.178(0.013) (0.010) (0.232) (0.025) (0.015) (0.852) (0.021) (0.009) (0.671)Black -0.092 0.134* 3.475 -0.213 0.084 2.586 -0.161 0.145 1.873(0.113) (0.065) (0.073) (0.151) (0.094) (0.123) (0.230) (0.111) (0.185)Const. -1.422*** -1.533*** 0.313 -1.835*** -1.678*** 0.444 -1.874*** -1.788*** 0.086(0.161) (0.084) (0.585) (0.193) (0.101) (0.513) (0.197) (0.104) (0.781)F Statistics 18.063 17.658 1.103 8.716 12.381 1.108 8.719 8.621 1.104Obs. 3399 3399 3399 2562 2562 2562 2562 2562 2562Notes: The Objective columns list the probit estimation results using the realization of the event as the dependentvariable. The Subjective columns lit the fractional probit estimation results using the subjective probability as thedependent variables. The Test columns list the F-value of testing the equality of the proceeding two estimates.Standard errors are listed in the parenthesis in objective and subjective columns, and p values are listed in theparenthesis in test columns. A single asterisk denotes significance at the 10% level, double for 5%, and triple for1%.793.4. Age Bias and Income BiasTable 3.4: Main Results: Female SampleWave 6 Wave 5 Wave 4Objective Subjective Test Objective Subjective Test Objective Subjective TestAge 0.043*** 0.022*** 24.144*** 0.053*** 0.027*** 25.895*** 0.054*** 0.027*** 38.679 ***(0.006) (0.003) (0.002) (0.004) (0.005) (0.004) (0.001) (0.004) (0.001)Income -0.242* 0.033 6.124* -0.002 0.070* 1.126 -0.091 0.044 1.754(0.103) (0.034) (0.021) (0.061) (0.037) (0.295) (0.087) (0.055) (0.195)Years of Schooling 0.013 0.036*** 2.276 0.025 0.017* 0.065 0.021 0.024*** 0.586(0.011) (0.012) (0.144) (0.016) (0.010) (0.805) (0.013) (0.013) (0.457)Health Condition 0.061** 0.062*** 0.003 0.093*** 0.063*** 1.574 0.094*** 0.076*** 0.204(0.022) (0.013) (0.963) (0.022) (0.012) (0.222) (0.023) (0.024) (0.666)ADLs 0.101*** 0.022 7.505** 0.084** 0.044* 2.055 0.075* 0.054** 0.153(0.022) (0.025) (0.013) (0.024) (0.026) (0.164) (0.034) (0.025) (0.704)IADLs 0.065* 0.095*** 0.976 0.070* 0.050* 0.490 0.044 0.075*** 0.527(0.038) (0.028) (0.337) (0.037) (0.028) (0.498) (0.034) (0.025) (0.476)Nursing Home Expe. 0.474*** 0.066 6.045* 0.495** 0.017 8.468** 0.456** 0.210* 1.444(0.135) (0.116) (0.026) (0.153) (0.108) (0.015) (0.145) (0.088) (0.245)LTC Insurance 0.112 0.123** 0.021 0.116 0.223*** 2.706 0.066 0.169** 0.848(0.073) (0.044) (0.895) (0.060) (0.040) (0.116) (0.096) (0.067) (0.369)Cohabitation -0.100 -0.014 2.428 -0.156** -0.041 4.806* -0.125* -0.028 2.313(0.053) (0.034) (0.133) (0.043) (0.037) (0.032) (0.054) (0.035) (0.148)Children -0.013 -0.018 0.056 -0.014 -0.017 0.148 -0.026 -0.028* 0.006(0.012) (0.014) (0.831) (0.015) (0.011) (0.712) (0.017) (0.013) (0.977)Black -0.128 -0.028 1.301 -0.159 0.046 2.916 -0.323** -0.155* 2.284(0.085) (0.056) (0.264) (0.102) (0.054) (0.091) (0.104) (0.069) (0.145)Const. -1.695*** -1.783*** 0.476 -1.925*** -1.557*** 3.154 -1.938*** -1.755*** 0.767(0.183) (0.084) (0.024) (0.194) (0.087) (0.085) (0.186) (0.096) (0.396)F Statistics 28.920 33.204 1.051 30.049 31.537 0.459 27.303 29.632 1.216Obs. 4709 4709 4709 4617 4617 4617 4583 4583 4583Notes: The Objective columns list the probit results using the realization of the event as the dependent variable. TheSubjective columns lit the fractional probit results using the subjective probability as the dependent variables. TheTest columns list the F-value of testing the equality of the proceeding two estimates. Standard errors are listed in theparenthesis in objective and subjective columns, and p values are listed in the parenthesis in test columns. A singleasterisk denotes significance at the 10% level, double for 5%, and triple for 1%.803.4. Age Bias and Income Biasholding a long term care insurance policy only have mild effects. If mostof the male seniors are enrolled in the policy through employee group pur-chasing, the mild effect seems reasonable. Though it is popularly believedthat male seniors living alone have greater demand for long term care, thecoefficient of cohabitation shows it is not so true after controlling for otherconfounding variables. Having more children does not have a strong effect.Most of the estimates have the sign we expected in the descriptive analysis.Second, the estimates are very close across the three waves. The age coeffi-cients are almost identical across the waves. This could be a good sign forthe specification and the selection of the covariates.Column 2 lists the estimates from the fractional probit estimator wheresubjective probability is the dependent variable for wave 6, while column5 for wave 5 and column 8 for wave 4. Again, standard errors are listedin the parentheses. Some interesting features deserve emphasis. First, al-most all estimates have the same sign as that of the corresponding ones inthe objective probability regressions. Second, the estimates across the threecolumns are generally close. This is obvious for the estimates for age, healthcondition and LTC insurance policy holding. The stable pattern supportsthe conclusion that self-reported subjective probability can be seriously an-alyzed. Column 3 lists the F -value from testing the hypothesis that thecoefficients of the corresponding covariate in column 1 and column 2 areequal. Similarly, column 6 for the coefficients in column 4 and 5, and col-umn 9 for the coefficients in column 7and 8. Here listed in the parenthesesare the p-values.The first row shows the results of the age covariate. In both the objectiveand subjective regressions, the estimates are precisely estimated and are veryclose. And as a result, the equivalence hypothesis is rejected only in wave 6,but not in other waves. Since age is highly correlated with other covariatesand unobservables, the results shows that the male seniors actually did wellin internalizing age information into their subjective expectation.The second row shows that the household income does not predict sub-jective probability as it does objective probability in terms of both magni-tude and significance level. It suggests male respondents do not correctly813.4. Age Bias and Income Biasinternalize household income information. The subjective estimates havethe opposite signs to the objective ones, and the hypothesis of zero cannotbe rejected. However, income does predict the objective chances very well.The F -values of the equivalence hypothesis test are 5.83 for wave 6, 5.97 forwave 5, and 5.18 for wave 4, all of which lead to a rejection of the hypothesisat a significance level less than 5%. The significantly negative estimates ofthe subjective income coefficients could be a result of the negative selectionresulting from Medicaid, which requires seniors to satisfy the income limitfor qualifying for Medicaid eligibility. Alternatively, it could be that lowerincome seniors are generally less healthy and thus have a higher demand fornursing home care. It could also be that higher income seniors have morelong term care options, like home health care and informal care from familymembers. Thus, the income bias can arise either because of the lack of Med-icaid eligibility information or a misunderstanding of the effect of income.To further understand the Medicaid effect on nursing home utilization, wealso controlled for non-housing assets and/or other assets measurement, butthe results are not sensitive to these and therefore are omitted.The third row gives the estimates of years of schooling. The estimatesin the objective probability regressions indicate that schooling years is nota powerful predictor of the chance of entering a nursing home, while theestimates in the subjective probability regression show that seniors slightlymisunderstand it, e.g., the subjective coefficient in wave 6 is 0.025(0.003),which is much higher than the corresponding objective coefficient. Overall,the equivalence hypothesis is only rejected at wave 6 with a significance levelof 10%. Since schooling years is generally believed to be highly correlatedwith other unobservable ability or cognitive factors, the results can be con-sidered as a sign indicating no serious measurement error in the subjectiveprobability. Both age and education information are correctly internalizedand the hypotheses are usually not rejected, but the underlying explanationscan be different. Age is a significant predictor, while education is not. Itshould be an easier task to abandon some less significant information whileforming expectation. To correctly use a significant predictor demands muchmore rationality and cognitive cost: the equivalence of the age coefficients823.4. Age Bias and Income Biasrepresents a precise process of collecting, assessing, judging and applying in-formation. Thus, rows 1 and 2 suggest the male seniors are quite successfulin tagging important information and applying it in forming expectations.The fourth row indicates that the seniors utilize health conditions verywell: the magnitude of the estimates in both objective and subjective proba-bility regressions are close enough that a F -test of the equivalence hypothesiscannot be rejected at any reasonable significance level.The fifth row gives the estimates for the ADLs covariate. The esti-mates and tests reveal some mild evidence for optimism bias: the size ofthe estimates, though generally significant, is much smaller than that of theestimates in the objective probability function. It seems male seniors actu-ally take less account of the ADLs in projecting their subjective probabilitythan its true weight in forming objective probabilities. In other words, theseniors knew the value of the information included in ADLs, but somehowunderestimate its magnitude. However, the optimism is only mild. Thehypothesis is only rejected in wave 5 because, in other waves, the objectivecoefficients are not precisely estimated.The sixth row gives the estimate for IADLs. The estimates are verysmall relative to the standard errors and the equivalence hypothesis can notbe rejected. The estimate of past nursing home stay experience is given inseventh row. As we have discussed, after controlling for other characteristics,the past nursing home experience actually does not significantly predict thechance.The eighth row is especially interesting. The objective coefficients showthat those with long term care policies do not have a significantly higher riskthan those without. However, the subjective coefficients are all significantat 1% level at all waves. In other words, those male seniors with a policyactually significantly reported a higher subjective probability, even aftercontrolling for other characteristics. As we mentioned, most of the policyholders are enrolled through workplace benefit. This result reveals someimportant potential bias though the hypotheses are only rejected in wave 4.Once again, the non-rejections merely reflect wide dispersion of the objectivecoefficients. If the sample excluded those purchasing policies in the retail833.4. Age Bias and Income Biasmarket, the equivalence hypotheses are rejected. We leave it to furtherresearch to discuss how bias can effect health insurance market outcomes.The next three rows show that cohabitation, quantity of children andrace do not affect the chance of entering a nursing home nor the subjectiveprobability of entering a nursing home. The last row gives the results for theconstant term. In all the cases, the equivalence hypothesis can?t be rejected.Table 3.3 thus presents some strong evidence the expectation bias interms of household income and some mild evidence supporting the biases ofage, ADLs and schooling. The conclusion should be read cautiously, how-ever. Any identified expectation bias can actually be a result of unobservablefactors excluded from our specification, which is discussed in the last partof section 3.2.2.The results for the female sample are listed in Table 3.4, which hasthe same structure as Table 3.3. Columns 1, 4 and 7 list the estimatesof the objective probability regression and column 2, 5 and 8 correspondto the estimates of the subjective probability regression. Though Table3.4 shares some features with Table 3.3, it differs in the following points.First, the estimated objective coefficients of the covariate age are muchhigher than the corresponding coefficients for the male sample, which impliesthat age actually has a stronger effect on females than on males. However,the subjective coefficients for females are almost identical with that of themale sample. As a result, we find very strong age bias for female seniorsacross all waves. All of the equivalence tests are rejected at a level less than0.1%. Since age is one of the easiest factors to assess and is informative toall seniors, the female age bias represents serious optimism among femaleseniors. Of course, some possible threats to the identified age bias do exist: itcould be the older seniors do not correctly express the subjective probabilitybecause of wishful thinking.Second, contrary to the male result, household income does not signifi-cantly affect the chance of nursing home utilization. The subjective coeffi-cients vary from year to year and are not significant. This could be a resultof the lack of response to income information while calculating the subjec-tive probability. Over all, no robust income bias was found among female843.4. Age Bias and Income Biasseniors.Third, past nursing home experience has better predictive power forfemales than for males. In all the estimations, the objective coefficients areabout 0.45 and highly significant, but the subjective coefficients are not ofsimilar magnitude and the tests of equivalence are mostly rejected. It seemsthere is significant optimism among female seniors. Notice the female seniorshave the same pattern as the male seniors regarding the long term insurancestatus.Combining the evidences from the two tables, we conclude a significantage bias among female seniors and income bias among male seniors 6. Biasesin terms of ADLs, LTC insurance status and past nursing home experienceare only mild. Understanding the true source of bias and gender differencescan help us make some progress in understanding how people form expec-tations. In the remainder of the paper, we consider some potential sourcesof the biases after examining several threats to the main results.3.4.2 Threats to the Main ResultsTwo major threats could invalidate the discussed biases. One is the vio-lation of the identification assumption that is the weak orthogonality con-dition. The long term nursing home care industry in the US was highlyregulated and various public policies were proposed and implemented inthe last decades. Different states adopted different approaches to regulatelong term nursing home care, but commonalities between jurisdictions al-low us to evaluate if the regulatory regime is truly a threat to the results.Another major threat is the inconsistent estimate of ?o resulting from thenon-classical measurement error of subjective probability, which could becaused by cognitive factors or wishful thinking by the seniors.6The gender bias difference is an interesting question. Though many potential reasonscan be drawn by daily observation on gender difference, one of particular interest is thatfemale?s own income might be less important, but age could be more relevant in marriageand labour markets.853.4. Age Bias and Income BiasPublic Regulation PolicyUnlike many other health care markets, the nursing home market was notcompletely competitive because of supply regulation policies. Many statesfeatured a moratorium on new entry or expansion; beds would not be addedwithout a certificate of need (CON). A nursing home provider could not enterthe market or add new beds without demonstrating the need for more beds,either by the occupancy rate or the ratio of bed to aged population. It isreported that the percentage of applications denied of all CON applicationswas about 40% in 1990s (Harrington et al. [1997]). In such an extensivelyregulated market, it is natural to suspect that regulatory policies wouldinclude some subjectively unexpected residual information correlating withthe demographic and social economical profiles of seniors. However, scruti-nizing the regulation policies during our study periods shows this worry islikely unfounded.First, the CON?s legitimacy is granted by the National Health Planningand Resources Development Act of 1974 (Public Law 93-64). The goals ofthe act are to improve the health of residents, to increase the accessibilityof quality of health service, to contain health care costs, and to preventunnecessary duplication of health services. To achieve these goals, the actrequired state approval for all new construction or expansion of health fa-cilities. States quickly realized CONs are a cost containment strategy tolimit rapidly increasing Medicaid expenditure. The CON regulation shouldbe within the framework of Public Law 93-64, which is openly available tothe market and consumers alike. In other words, it is unlikely there is anysubjectively unexpected part of the policies that would be systematicallycorrelated with the market characteristics.Second, though states implementing CON vary over years, it was stableduring our study period except for Indiana, which repealed the CON lawin 1999 again after reinstating it in 1997 following an initial sunset of thelaw in 1996. Note, survey wave 4 was conducted in 1998, wave 5 in 2000and wave 6 in 2002. The fluid situation in Indiana might partially explainthe observed rejection of the equivalence hypothesis of the constant term in863.4. Age Bias and Income Biaswave 6 of male sample, but should has no impact on our major results. Inall these years, though amendments are frequently made to the CON laws,there are no major amendments concerning the regulation of the nursinghome facility.Given these observations, it seems fair to claim that the policies regulat-ing the LTC market during our study period do not include any subjectivelyunexpected information that is correlated with the controlled observables inthe market.Measurement ErrorAnother major threat is the possibility of inconsistent estimation of ?o onthe subjective side due to the measurement error of subjective probability.Though there are potentially many factors affecting the measurement error,the cognitive factors are the major concern since estimating subjective prob-ability is a process of applying cognitive ability to formulate a quantitativeresult.This section tests whether there is non-classical measurement error dueto cognitive ability in the variable of subjective probability. Our methodexploits the assumption that if cognitive factors undesirably affect the mea-surement of the probability of entering a nursing home, it should affectmeasurement error of other subjective probabilities. Consider the reportedsubjective probability ps = p?s + ?, where p?s is the true subjective proba-bility and ? is the measurement error.The measurement error ? includes allcognitive factors that influence how people report the subjective probabil-ity ps. Though ? is unobservable, we have encountered its proxy earlier inour discussion. In the warm-up exercise(See discussion in section 3.3.1), therespondents were asked to assess the chance of raining tomorrow. Similarly,the reported probability of rain can be decomposed as the true probabilityand the measurement error. Since the true probability of raining is random,it should be uncorrelated with other subjective probability judgments. Thus,regressing the subjective nursing home probability on the reported proba-bility of rain would generate a zero coefficient in the absence of systematic873.4. Age Bias and Income BiasTable 3.5: Assessing the Measurement Error due to Cognitive AbilityWaves 4 5 6The Probability of Raining .00012(.134) -.00005(.585) -.00004(.595)R2 .0003 .0001 .0000N 9359 9568 10118Notes: The dependent variable is the subjective probability of entering anursing home. A single asterisk denotes significance at the 10% level, doublefor 5%, and triple for 1%.effects of cognitive factors. Alternatively, if the zero hypothesis were re-jected, cognitive factors may shape the measurement in an undesirable way.Table 3.5 lists the regression estimates for the survey waves from 4 to6. All of the estimated coefficients are very small and not significant. Theextremely low values of R2 support the claim as well. Thus, the evidencesuggests that the measurement error due to cognitive factors cannot be adriving force of the identified biases.Another interesting factor affecting measurement error is wishful think-ing. Though the definition varies from case to case, it has been documentedin many fields. Babad and Katz [1991] found soccer fans over estimated thechance of victory for the side they favoured. Gordon et al. [2005] showedin an experimental environment that people selectively remember and for-get to bias toward the information sources that are consistent with desiredoutcomes.Directly assessing the impact of wishing thinking on measurement erroris difficult since the later is unobservable. Alternatively, we assess the im-pact of wishful thinking on female age bias by utilizing an implication ofwishful thinking. The wishful thinking argument notices the age bias aremainly driven by the smaller coefficients of age in the subjective probabil-ity equations and suspects the subjective probability measurement error ispositively correlated with age, that is the older respondents intentionallyreport a lower subjective probability because of wishful thinking 7. If this7Of course, lower should be read as lower than the true subjective probability.883.4. Age Bias and Income BiasTable 3.6: Is Age Bias a Result of Wishful Thinking?Panel A: Compare the Age Estimates By Age GroupsWave 6 Wave 5 Wave 4Older Younger Older Younger Older YoungerAge 0.01** 0.02** 0.02** 0.02* 0.03*** 0.02(0.00) (0.01) (0.01) (0.01) (0.01) (0.01)F Statistics 28.92 32.15 23.97 23.99 23.54 31.45Obs. 2270 2441 1732 1816 1247 1503Panel B: The Pattern of Age CoefficientsWave 6 Wave 5 Wave 4Age 0.01** 0.02** 0.02**(0.00) (0.00) (0.00)F Statistics 33.25 36.52 31.31Obs. 2761 2760 2747Notes: Panel A lists the age estimates for the age group above the agemedian and the group below the age median of the subjective probabilityregression. Coefficients of other controls are omitted. Panel B lists the agecoefficients of the subjective probability regression for the same respon-dents at various survey years. Coefficients of other controls are omitted.A single asterisk denotes significance at the 10% level, double for 5%, andtriple for 1%.were true, the age bias would merely be a reported bias: the identified agebias is purely a result of the measurement error rather than not correctlyinternalizing information. To evaluate this statement, we divide the femalesample into two groups: those above median age and those below medianage groups. Should the theory hold, we will observe a smaller subjectivecoefficient in the above median age group.Panel A of Table 3.6 presents the estimation results. Obviously, theestimated coefficient for the younger seniors is actually smaller than theolder sample in wave 4, equal in wave 5, though is lower than that of olderseniors in wave 6. The evidence implies that the age bias is not likely due towishful thinking or intentionally underreporting of the subjective probabilityby the seniors.As a complementary analysis, we also examine the age coefficients of893.5. Sources of Biasesthe identical respondents at different ages. More concretely, the methodcompares the age coefficients of the subjective probability regression for thesame respondents in various survey years. The various ages are availablebecause of the panel structure of the data set. By restricting the sampleavailable in all three waves, the data set ends up with 2761 respondents.The intuition here is also from the implication of wishful thinking: if theseniors did report a lower subjective probability while becoming older, theage coefficients should decrease from wave 4 to wave 6. In panel B of Table3.6, though the estimated age coefficient does decrease from wave 5 to wave6, the decrease is not observed from wave 4 to wave 5. The lack of a consis-tent pattern of the age estimations thus weighs against the wishful thinkingconjecture.3.5 Sources of BiasesThe finding of age and income biases brings the interesting question aboutthe root causes of the biases. A root cause or source is any factor that(i) influences the assessment of subjective probability but has no effect onobjective probability and (ii) the hypothesis of equivalence of ?o and ?? ofthe biased covariate could not be rejected after controlling for the sourcein subjective regression. Consider the age bias of female seniors in theabove analysis. In all three waves, these seniors assign a lower weight onage information in forming their subjective assessment. Various reasonscan potentially cause the departure of subjective assessment from the right,objective weight. For example, if cognitive ability is the source of age bias, itshould not affect the objective probability but increases age coefficient in thesubjective regression so that the equivalence hypothesis of age coefficientswould NOT be rejected if cognitive ability is controlled for in the subjectiveregression.Any potential sources of bias should first be tested against condition (i),which especially requires exclusion of a potential source from objective prob-ability. Two potential bias sources, cognitive ability and a misconceptionbetween risk aversion and probability, are believed to satisfy the condition.903.5. Sources of BiasesWhile it is feasible to validate that both factors have substantial impacton subjective assessment, verifying or refuting the exclusion requirement ismore or less subjective. By simply maintaining the exclusion assumption,the next subsections are devoted to demonstrating both potential sources?impact on subjective probability assessment and reporting the empirical re-sults of hypothesis tests.3.5.1 Cognitive AbilityIt has been long known in the social sciences that subjective probability as-sessment depends on cognitive ability. Besides the seminal work of Tverskyand Kahneman [1971, 1974, 1981, 1985], Hogarth [1975] formally discussedthe implication of cognitive process for the assessments of subjective prob-ability. He argued assessing probability is a ?selective, sequential informa-tion processing system with limited capacity?. Wallsten and Budescu [1983]wrote:Upon being asked to evaluate the probability of an outcome,a person will search his or her memory for relevant knowledge,combine it with the information at hand, and (presumably) pro-vide the best judgment possible. That judgment will depend onwhat is retrieved from memory, what aspects of the current in-formation are utilized, and possibly on the sequential order inwhich this is all integrated into a unified opinion.Clearly, cognitive ability is believed to have a central role in the process ofsubjective probability assessment.On the other hand side, age differences in various measures of cognitivefunction have often been reported(see Salthouse and Mitchell [1990] andHertzog [1989]). These papers generally found that age differences are sub-stantial and a considerable proportion of age related variance can be tracedto cognitive ability limits.These readings thus imply a potential bias source: the identified biases,especially age bias, might be attributed to cognitive ability limits in for-mulating the subjective probability. Fortunately, various measurements of913.5. Sources of BiasesTable 3.7: Are Cognitive Factors the Source of Biases?Panel A: Age Coefficient, Females SampleWave 6 Wave 5 Wave 4Objective Subjective Test Objective Subjective Test Objective Subjective TestAge 0.04*** 0.02*** 19.95*** 0.05*** 0.02*** 15.90*** 0.06*** 0.02*** 15.79***(0.00) (0.00) (0.00) (0.01) (0.00) (0.00) (0.01) (0.00) (0.00)F Statistics 28.92 25.62 1.97 23.99 31.02 1.97 13.44 24.97 1.97Obs. 4709 4709 4709 3545 3545 3545 2747 2747 2747Panel B: Income Coefficient, Male SampleWave 6 Wave 5 Wave 4Objective Subjective Test Objective Subjective Test Objective Subjective TestIncome -0.25* -0.01 5.77* -0.18* 0.02 6.01* -0.26* 0.00 4.57*(0.09) (0.02) (0.02) (0.08) (0.03) (0.02) (0.12) (0.04) (0.03)F Statistics 18.06 19.35 1.97 8.71 7.61 1.97 4.33 6.95 1.97Obs. 3399 3399 3399 2562 2562 2562 2012 2012 2012Notes: This table lists the similar results as the main result table except the cognitive factors are also includedin the subjective columns, the estimates of which and other controls are not listed. A single asterisk denotessignificance at the 10% level, double for 5%, and triple for 1%.cognitive ability are included in the HRS data set and thus can be formallycontrolled for in our regression. Briefly, these measurements include mem-ory, mental status, vocabulary for most of the respondents, and numericaland quantitative reasoning for a small sample. The memory functioningmeasurements include self-reported memory, immediate word recall tests,and delayed word recall test. The mental status is measured by varioustests that assess knowledge, language and orientation. These measurementsinclude the serial seven test, backwards counting, date naming, object nam-ing, and naming the President and Vice President of the United States.Because of a very small sample for the quantitative test, we disregard it inthe analysis.Table 3.7 lists the results after controlling for the above cognitive factorsin the subjective probability function. Doing so does not change the result.In panel A, the F -value of the hypothesis test in wave 6 decreases from about24.14 to 19.95, but the changes do not alter the identification of age bias.923.5. Sources of BiasesThe inclusion of the cognitive ability factors also slightly decrease most theF -values, but not the significance level. Since cognitive factors could alsobe correlated with income, panel B presents the results after controllingfor cognitive factors in the subjective regression equation for male sample.Similarly, controlling for the cognitive factors only has a minor effects onthe estimation. The evidence presented here thus indicates that althoughcognitive factors do have some minor effects on age bias and income bias,they are not the driving sources of the observed biases.3.5.2 Risk AversionAnother potential source of biases is the difficulty in separating risk aversionfrom subjective probability, i.e., the reported probability somehow measuresrisk aversion as well. If so, the reported subjective probability contains someinformation about personal risk preference. Many articles have reported thatage and other individual characteristics are correlated with risk aversion. Forexample, by studying the relationship between age and risky asset holding ofCanadian households, Morin and Suarez [1983] concluded that risk aversionincreases with age. Sung and Hanna [1996] analyzed the response to risktolerance questions and weakly conclude that risk tolerance decreases withage. However, Wang and Hanna [1997] reported that risk tolerance actuallyincrease with age after controlling for other individual characteristics. Onthe other hand side, correlation between income and risk aversion has alsobeen reported. Shaw [1996] argued that risk aversion can affect humancapital investment. By developing a model of joint investment in financialwealth and human wealth, Shaw showed that human capital investment isan inverse function of risk aversion and reported empirical evidence for thepositive correlation between wage growth and risk taking.This section develops a measure to proxy for risk aversion using theinformation available in the HRS and test if risk aversion could be a sourceof bias. Various methods have been proposed to measure risk aversion inthe HRS data set. First, the ratio of risky assets to total assets could bea measure of risk aversion. The risky asset is defined as the net value of933.6. Conclusionstocks, mutual funds and investment trusts. Using the actual outcome data,we can directly measure the interesting parameter and thus minimize themeasurement error. The disadvantage is that the directly observed outcomesare also an equilibrium result and thus include the idiosyncratic shocks andconstraints. Alternatively, the income risk aversion test result measures riskaversion as well. In the risk aversion test, the respondents are asked to choosebetween a pair of jobs where one guarantees the current family income andthe other one offers an even chance to increase income but also the chanceto lose income 8. The major disadvantage of responsive risk aversion is itssubjective nature: its measurement is influenced by the same factors thatinfluence subjective probability. The measurement also has only a smallsample since it was only asked in wave 4. Considering all the benefits andcosts, we thus focus on the behavioural measurement of risk aversion, thatis the ratio of risky assets to total assets.Table 3.8 shows the results of estimation after controlling for risk aversionin the subjective probability regression equation. Though doing so doesaffect the estimated coefficients, it doesn?t change the general pattern ofeither age bias or income bias.3.6 ConclusionRecent academic consensus believes that incorporating subjective expec-tations into economic analysis of various decision making process can effec-tively improve prediction efficiency. To do so requires a better understandingof expectations formation.This paper contributes to increasing academic interest in expectationformation by defining expectation bias and presenting a unified strategy ofidentification. The first innovation in our method is to exploit the impli-cation of rational expectations, which says people form objectively correctexpectations conditional on what information they possess. In the static8More precisely, the risk aversion is measured at four levels by asking if the respondentwould take a job with even chance of doubling income or cutting it by a half, a third ora fifth. The most risk aversion is defined as not accepting any of the above chance andchoose the status quo.943.6. ConclusionTable 3.8: Is Risk Aversion the Source of Biases?Panel A: Age Coefficient, Females SampleWave 6 Wave 5 Wave 4Objective Subjective Test Objective Subjective Test Objective Subjective TestAge 0.04*** 0.02*** 22.90*** 0.05*** 0.02*** 13.82*** 0.06*** 0.03*** 15.43***0.00) (0.00) (0.00) (0.01) (0.00) (0.00) (0.01) (0.00) (0.00)F Statistics 28.92 25.64 2.32 23.99 21.33 2.32 13.44 17.48 2.32Obs. 4709 4299 4709 3545 3227 3227 2747 2519 2519Panel B: Income Coefficient, Male SampleWave 6 Wave 5 Wave 4Objective Subjective Test Objective Subjective Test Objective Subjective TestIncome -0.25* -0.01 5.80* -0.18* 0.02 5.95* -0.26* 0.02 5.08*(0.09) (0.02) (0.02) (0.08) (0.03) (0.02) (0.12) (0.05) (0.02)F Statistics 18.06 16.77 2.32 8.71 6.58 2.32 4.33 4.91 2.32Obs. 3399 3159 3159 2562 2403 2403 2012 1893 1893Notes: This table lists the similar results as the main result table except the risk aversion is also included in thesubjective columns, the estimates of which and other controls are not listed. A single asterisk denotes significanceat the 10% level, double for 5%, and triple for 1%.context, rationality is equivalent to the statement that the information rep-resented by some related covariates should be the same weight in predictingboth the objective and subjective probability. Following this intuition, wedefine expectation bias as the inequality of the coefficients of a covariate inobjective and subjective probability equations.Since objective probability is usually unavailable, identifying bias there-fore requires a proxy for it. In our analysis, we substitute the objectiveprobability with the ex post realization of the interesting event, which intro-duces interim factors to influence the realization. However, our discussionshows that as long as the interim events satisfy a orthogonality condition,substitution does not contaminate the identification strategy.Our empirical analysis using the Health and Retirement Study finds pri-mary evidence supporting expectation biases among seniors. More specif-ically, we find female seniors fail at correctly integrating age informationwhen they try to formulate their subjective probability of entering a nursing953.6. Conclusionhome within five years. A similar result is also found for male seniors wherethey fail to correctly internalize income information. Robustness analysissuggests the biases are not driven either by measurement error in subjectiveprobabilities or a possible intermediate event, e.g., public policies regulatingnursing homes.Though cognitive ability has a minor effect on identified bias, it is not themajor source. Neither is risk aversion. Our result presents a challenge to theconventional practice of assuming objectively correct expectations. Thoughit has been recently discussed in various experimental environments, ourpaper relies on real data, rather than observations in a lab environment.The generality of the result means that it can be potentially applied tounderstand various consumer behaviours under uncertainty.96Chapter 4Risk Aversion vs. BequestMotive: How do SeniorsMake Long Term CareInsurance Decisions?4.1 IntroductionUnderstanding how seniors make long term care insurance (LTCI) purchas-ing decisions is of importance for both policy makers and researchers. Inthe USA, long term care (LTC) expense is the single largest financial riskfacings seniors with an annual cost of more than $200 billion. Yet, onlyabout 10% of seniors are covered by any LTCI policies. According to classi-cal asymmetric information theory, the LTCI market is expected to observea positive correlation between insurance coverage and incident occurrencedue to adverse selection and moral hazard (See Fang, Keane and Silverman[2007] for a concise discussion). However, recent research has noticed that inthe LTCI market those covered by an insurance policy do not experience ahigher incidence rate than those who do not (See Finkelstein and MaGarry[2006] for a thorough discussion).Many institutions and market factors might help to understand the issuesof thin market size and the absence of positive correlation. Preferences, e.g.risk aversion and bequest motive, have been recently proposed as drivingforces to explain these issues. For example, Pauly [1990] analyzed potentialreasons for the small LTCI market size by arguing that a LTCI policy cov-974.1. Introductionerage?s primary objective is to protect bequest, which are already excessivebecause of imperfect annuity markets. Regarding the absence of a positivecorrelation in the LTCI market, one explanation is that seniors vary in theirrisk aversion (see Finkelstein and MaGarry [2006]), in addition to exogenousrisk status. In other words, it relies on a negative correlation between riskaversion and risk status to offset the potential positive correlation.Naturally, risk aversion as well as risk status should play important rolesin driving seniors? LTCI shopping decisions. Just as any other insurancepurchasing decisions, a rational individual needs to evaluate her/his riskstatus (e.g. high vs. low) and decide to buy or not based on idiosyncraticrisk preference. On the other hand, because of both the risk feature of longterm care and practice of Medicaid policy, bequest motive would be an otherpotential driving force. For American seniors, the life time chance of utilizinglong term care is substantial, which is estimated at more than 40%. Thoughthe Medicaid program pays most of the LTC expenses, its estate recoverypolicy influences the decisions of seniors because the program actually seeksto recover all of the expenses it paid for a senior from estate upon the deathof the senior.Integrating a bequest motive into the model could facilitate better un-derstanding of many phenomena related to seniors? consumption, savingand investment etc.. For example, Vidal-Melia? and Leja?rraga-Garc??a [2006]tried to understand the annuities puzzle by introducing a bequest motivebut failed to find it a significant factor influencing the demand for annuitiesat all. Specially, Kopczuk and Lupton [2007] examined the effect of observedand unobserved heterogeneity in the desire to die with positive net worth.They found that roughly three-fourths of the elderly single population has abequest motive and concluded that among elderly single households, aboutfour-fifths of their net wealth will be bequeathed and approximately half ofthis is due to a bequest motive. More recently, Lockwood [2011] disentangledprecautionary and bequest motives by exploiting LTCI purchasing decisionsand saving patterns across wealth distribution and found bequest motivesvery important in explaining dissaving behaviour and the small demand sizeof the LTCI market.984.1. IntroductionBased on these evidence, it seems promising to view a senior?s LTCIdecision by integrating both risk aversion and a bequest motive. To do so,this chapter develops a dynamic structural discrete choice model where arational, risk averse and bequest motivated single senior has to decide ateach period whether to buy a LTCI policy or not. In this model, bothrisk aversion and a bequest motive determine a senior?s value function andtherefore drive the senior?s LTCI decision. By carefully constructing howseniors make insurance and consumption decisions, the chapter is capableof estimating a joint distribution of risk aversion and bequest motive fromobserved LTCI choices.This chapter gives insights to many related debates over the LTCI marketand Medicaid policy reform because of its estimate of primitive parameters,which are these directly describing preferences. First, it helps to resolvethe positive correlation puzzle observed in LTCI market. Fang, Keane andSilerman [2008] discussed the advantageous selection in the Medigap insur-ance market, that is those with the Medigap coverage actually spend lessthan these without conditional on controls for the Medigap price, and con-cluded that cognitive ability is the sources of advantageous selection. Ourpaper complements their results in the LTCI market and finds that a be-quest motive, rather than risk aversion, drives the absence. Second, thoughthis chapter could not answer the thin market size issue directly, it doesoffer answers to a series of questions about how policy changes can affectthe LTCI market size since our model captures relevant institutional details.This rest of this article is organized as follows. In section 4.2, we presentsome introduction to Medicaid policy, especially its spending down and es-tate recovery policy. Since LTCI eligibility shapes our sample selection, itis also briefly discussed. Section 4.3 describes the data source and pro-vides some primitive analysis on how seniors make LTCI decisions. Section4.4 first provides a description on the structural model being followed by adiscussion on identification. The identification depends on a monotonicityproperty of the structural model. Treatment on subjective beliefs and aparameter calibration method are also specified. The discussion ends with adescription of the likelihood function and empirical estimation method. Sec-994.2. Institution Backgroundtion 4.5 presents our estimate of the joint distribution of risk aversion andbequest motive. We find substantial heterogeneity in bequest motive. Theestimated risk aversion is quite homogeneous. By inferring each senior?sidiosyncratic preferences ex post, we also find bequest motive is a majorsource for the absence of positive correlation. This section ends with anal-yses on various counterfactual policy effects on LTCI market size. Section4.6 concludes.4.2 Institution Background4.2.1 The Eligibility Rules and Estate Recovery Policy ofMedicaidLong term care (LTC) 1 expense is the largest single health and financialrisk facing seniors in the USA. The total expenses reaches 135 billion dollarsin 2004 (CBO) and it is projected to double soon because of the retirementof the baby boom cohort. For individuals, the average annual nursing homecare expense jumped from $34, 156 in 1995 to $60, 249 in 2004 (Stewart etal. [2009]). And a senior?s lifetime chance of using long term care is morethan 40% (Kemper et al. [1991]).The funding of LTC expense comes from four sources. The majority isstate Medicaid programs, which pay for roughly half of nursing home ex-pense and about 70% of all bed-days. Another public insurance program,Medicare, pays about 20% of the expense, which is almost equivalent to theout-of-pocket payment. Private insurance accounts about 9% of the total ex-pense (GAO, [2005]). Unlike Medicaid, Medicare funding is predominantlyfor post-acute care of short-stays following hospitalization. Thus, for seniors,the major sources are Medicaid and out-of-pocket expense.However, Medicaid is a means-tested, needs-based social assistance pro-gram rather than a social insurance program. To build eligibility, individualsmust meet certain functional criteria as well as state-specified income and1Generally speaking, long term care includes nursing home care, home care and com-munity based care etc.. In this chapter, we do not distinguish these types of care andalways use long term care and nursing home care interchangably.1004.2. Institution Backgroundasset thresholds. The functional criteria generally require at least two ?ac-tivities of daily living? (ADLs) limitations, which include bathing, dressing,eating, toileting, transferring (from a bed to a chair or vice versa), andwalking across a room.The details of income and asset tests vary from state to state, but thereare some common features that should be taken into account when we modelseniors? insurance decisions. On the income side, if an individual?s incomeexceeds an income threshold published by the federal government, the indi-vidual is not eligible for Medicaid coverage. However, because of a spendingdown policy, or its counter part of Miller Trust 2 at some states, the indi-vidual could be eligible for Medicaid coverage after spending down his/hermonthly income below the threshold. In either cases, the individual has tospend his/her income to the threshold level before Medicaid intervenes andpays the remaining balance 3. On the asset side, the threshold varies a littlefrom state to state, and in most states it is $2000 dollars for a single withoutdependents during our study period. In calculating qualified assets, all cash,saving accounts, stocks, IRA and other retirement accounts are accountableassets. The major excluded asset is real estate. Thus, the asset test re-quires an individual to spend the entire balance of non-house assets to theexemption level before becoming eligible for Medicaid assistance. In sum,the means test of Medicaid requires a senior to spend down most incomeand non-house assets to some thresholds before Medicaid pays the shortfall.Besides the eligibility tests that could affect a senior?s private insurancepurchasing decision, the estate recovery policy of Medicaid is another rulethat should be included in our model. Upon the death of a senior who everbenefited from Medicaid for nursing home care, a state must, by law, seekto recover all of the expense it made on behalf of the senior. It is calledestate recovery because it is not executed until the death of the recipient.To avoid transferring assets by recipients upon death, states law puts pre-2Miller Trust, or qualified income trust, can be used to qualify a Medicaid applicantwith income in excess of the eligibility limit for long-term care assistance from Medicaid.3Of course, if a senior?s income is enough to cover all the monthly bill of nursing homecare, Medicaid will pay nothing. So, theoretically, seniors with above average income havea stronger incentive to buy insurance.1014.2. Institution Backgrounddeath liens on recipients? home and other assets. Though an estate recoveryis prohibited in certain instances when Federal law deems that the needsof certain relatives for estate assets take precedence over Medicaid claims,it is strictly executed if the recipient has no dependent relatives, whichinclude surviving spouse and dependent children. Combining the eligibilityand recovery policies, we can treat Medicaid as an interest-free need-cappedcredit lender without checking collateral.4.2.2 Long Term Care Insurance MarketDespite the large risk, private LTCI market is relatively small. Thoughprivate insurance policy covers about seven million lives in 2007 (Senkewicz[2009]), the coverage is only about 10 percent of all population aged over60. Unlike other acute health insurance markets where group purchasingbought most policies, almost 80% of LTCI policies are sold to individualsdirectly. The premiums charged for LTCI vary by the age when a policy wasinitially sold, with higher premiums charged to those bought at older ages.Thus, a senior will pay the same premium as what was paid at the first timeas long as the policy was never lapse, subject to the adjustment of inflationand other cohort size effect 4. Since the risk of using the benefit at youngerages is lower than that at older ages, the pricing of the LTC insurance isbelieved to reduce adverse selection at later stages. From the viewpoint of apotential policy buyer, the trade-off of buying today at a low premium rateor buying tomorrow at a higher rate always exists.Beyond the premium trade-off, a delay of buying decision also risks asenior losing approval of her/his insurance policy application. As reportedby McGarry and Finkelstein [2006], an LTCI application form generallyincludes detailed information on an applicant?s age, sex, and membership inone of seven different health states defined by the number of limitations toinstrumental activities of daily living (IADLs), the number of limitations toactivities of daily living (ADLs), and the presence of cognitive impairments4Adjustment to cohort size effect usually happens when risk incidence rate for a cohortis higher than expected by policy insurers and therefore the previous charged premiumsare not enough to cover cost.1024.3. Data and Primary Analysisetc.. Any positive answer to these questions results in the application beingdenied. Though the disapproval risk will not be explicitly modeled in ouranalysis, it has strong implications for sample selection: all these seniors whocould not get approved should be excluded from our analysis. Otherwise,inclusion of subjects who could not get covered by any private insurancepolicies will contaminate the inference of preferences.In sum, the Medicaid eligibility rules and recovery policy are two impor-tant factors in modeling LTCI purchasing decisions, while the risk of beingdenied in LTCI policy applications alarms caution in sample selection.4.3 Data and Primary AnalysisThis chapter uses the Health and Retirement Survey (HRS) 2002 data set.The HRS is a biennial, longitudinal survey of American beginning in 1992.The subjects surveyed in 2002 were also interviewed in 2004 which enablesus to observe the incidence of entering nursing home facilities. Like manyother surveys, the HRS includes very detailed wealth, health and insurancecoverage information. However, some unique features make it well suitedfor our study. It carefully designs and asks various subjective probabilitiesof entering a nursing home facility and living up to 75 years old etc.. Thosesubjective probabilities provide very rich private information about howforward looking seniors envision future events.The sample included in this chapter is seniors meeting the following de-mographic, health and wealth criteria: are single, between 60 and 70 andthus are covered or expect to be covered by Medicare policy, do not have anyADL or IADL disabilities, and have real estate assets more than $60, 000and yearly income at least $20000. To simplify the analysis, we also excludethose with negative liquid assets. We end with 696 observations. Undoubt-edly, the selected sample consists of quite healthy and wealthy single seniors.In order to exclude possible home care offered by cohabitants, the sample isrestricted to single seniors. As we discussed in subsection 4.2.2, the chanceof getting a LTCI approval is small for those with any ADL or IADL disabil-ities. With the same purpose of more precisely pinning down the preferences1034.3. Data and Primary Analysis(risk aversion and bequest motive), the threshold of housing assets is set tobe about one year of long term care cost. According to American HealthInsurance Plan [2007], about 95% of new buyers in year 2005 have an incomegreater than $20000. The highly selected sample would potentially estimatea less heterogeneous preference distribution. Alternatively, we could relaxthe selective criteria to have a more representative sample, e.g., removingthe real estate assets or income requirements. By including single seniorwith less real estate assets, it trends to have a smaller mean value but agreater standard error of bequest motive than otherwise. However, the im-pact of removing income requirement on the mean values of the distributionis not clear, though which would lead a more heterogeneous preference dis-tribution.Table 4.1 lists mean and standard deviation of various variables by LTCinsurance policy coverage status. Several features of the table are speciallyinteresting. First, incidence rates of nursing home care or risk status mea-sured in about five year horizon across the LTCI holding status are veryclose. Actually, the incidence rate is 0.046 for non-holders and 0.050 forpolicy holders and the hypothesis of identical incidence rates would notbe rejected. This is not surprising as the absence of positive correlationbetween LTC insurance coverage and incidence rate has been documentedin other research. Second, policy holders and non-holders are comparablewith respect to many other observable characteristics. Though policy hold-ers are marginally richer, and more likely to be female, they have almostthe same Body Massive Index (BMI) level, self reported health level andout-of-pocket medical expenditures as non-holders. Last, policy holders arevery close to non-holders in the risk aversion category, but have a higherlevel self-reported probability of leaving bequest and probability of enteringa nursing home facility in five years. Intuitively, were risk aversion a ma-jor factor driving seniors? LTCI decisions, we should observe its differenceacross LTCI holders and non-holders. Instead, what we observed are thedifferences in bequest motive and subjective incidence probability. We be-lieve an attempt to understand how seniors make LTC insurance purchasingdecisions should trace these facts.1044.3. Data and Primary AnalysisTable 4.1: Descriptive Statistics by LTC Insurance Purchase StatusNon Buyer BuyerIncidence Rate of Nursing Home 0.046 (0.209) 0.050 (0.219)Age 64.98 (3.19) 64.88 (3.13)Gender 0.29 (0.46) 0.17 (0.38)Income 54,777 (287,099) 78,962 (240,518)BMI 27.08 (5.04) 27.41 (4.87)Self Reported Health Level 2.48 (1.00) 2.37 (0.86)Out of Pocket Medical Expense 3,874 (17,239) 3,512 (7,506)House Value 163,555 (133,501) 207,200 (300,861)Risk Aversion Category 4.86 (1.45) 4.38 (1.60)Subjective Probability of Nursing Home Care 10.62 (18.89) 12.57 (21.11)Subjective Bequest Probability 51.46 (41.86) 64.66 (37.64)Observations 577 119Notes: Self Reported Health Level is coded with the range from 1 for Excellent to 5 for Poor.Risk Aversion Category is coded with the range from 1 for the least risk averse to 5 for the mostrisk averse in the hypothetical income game described in section 4.5.2. Subjective probabilityof nursing home care is the self-reported probability of entering a nursing home in five years.Subjective bequest probability is the self-reported probability of leaving any bequests. Standarddeviations are in parentheses.1054.3. Data and Primary AnalysisTable 4.1 presents some hints of understanding how seniors make LTCIdecisions. One way to resolve the puzzle is that individuals hold a seconddimension of private information which operates in opposite direction tooffset the positive correlation between risk status and policy coverage. Inthe table, though policy holders do not have a significant higher risk sta-tus (0.050 vs. 0.046), the holders do have some private information: thesubjective probability of leaving a bequest is higher than non holders. Italso suggests that risk aversion, the preference to risk, is unlikely to be thesecond dimension of information: the non holders actually have a slightlyhigher risk aversion level. In contrast, the subjective bequest probabilityshows a higher value among policy holders than among non-holders. Toform a better perspective, table 4.2 lists the average bequest probabilityand risk aversion level by insurance coverage and risk incidence status. Forexample, the average bequest probability for these not covered by any LTCinsurance and who have not incurred incidence is 51.79 with a standarddeviation 41.80. As we have seen above, the average bequest probabilityis higher for the policy holders than the non-holders and the average riskaversion is almost identical. Specially, The LTC insurance holders withoututilizing the policy actually have a bequest probability of 65.77 versus to44.56 for the non holders who actually entered a LTC facility. The similarcomparison of risk aversion is observed at 3.31 versus 3.67, which again istoo close to offset a positive correlation.Based on the primary evidence, we therefore propose that a bequestmotive, just like the already discussed risk aversion, is another major forcedriving LTCI decisions and might even play a more substantial role thanrisk aversion. However, the primary statistics are built on proxy variablesfor bequest motive and risk aversion. The next section develops a modelto infer the risk aversion and bequest motive distribution in order to betterunderstand how seniors make LTCI decisions.1064.4. Model: Specification, Identification and EstimationTable 4.2: Bequest Probability and Risk Aversion byInsurance Coverage and Risk Incidence StatusInsurance Coverage Buyers Non-buyersPanel A: Bequest ProbabilityRisk IncidenceYES 64.66(37.64) 44.56(43.22)NO 65.77(37.04) 51.79(41.80)Panel B: Risk AversionRisk IncidenceYES 3.66(0.81) 3.67(0.71)NO 3.31(1.07) 3.27(1.08)Notes: Standard deviations are in parentheses.4.4 Model: Specification, Identification andEstimationWe start by discussing a structural dynamic discrete choice model of a riskaverse and bequest motivated senior. The model captures the relevant prac-tice of Medicaid policy and the front-load pricing of the insurance policydiscussed above. Though a senior in our model optimally decides both thediscrete choice of buying a LTCI policy and the continuous consumptionlevel, our identification utilizes only the information from discrete choicesby exploiting a monotonicity property of the model. It does not fully ex-tract information from consumption choices since the observation is usuallypoor. See French and Jones [2010] for a different framework where momentconditions are employed to facilitate identification.Our identification hinges on the structure of the model, especially amonotonic relationship between LTCI decisions and preference parameters.During analysis, we first consider the case where individuals follow the mono-tonic decision rule: purchasing an insurance policy if and only if the value ofdoing so is greater than the value of not doing so, V b > V n. Similar mono-tonic decision rule method was applied in recent works of Cohen and Einav[2007] and Einav, Finkelstein and Schrimpf [2010] etc.. Besides its heavycomputation, it does not absorb any shocks and thus potentially provides1074.4. Model: Specification, Identification and Estimationa poor fit to observed data set. Thus, we later introduce an unobservabletype I extreme value distributed utility shocks following the work of Rust[1987]. In this scenario, the probability of purchasing an insurance policy isthus given by a logit probability model.4.4.1 SpecificationWe model the behavior of a rational, forward looking, risk averse, bequestmotivated, retired (or nearly retired) individual with an accumulated stockof liquid and housing asset, uncertainty about nursing home state and mor-tality risk, and time separable utility. A similar framework has been adaptedto model labour supply (Eckstein and Wolpin [1989]) and annuity choices(Einav et al. [2010]) etc.. In this model, a senior is assumed to maximizethe present value of utility over an unknown finite horizon by choosing con-sumption and whether or not to buy a long term care insurance policy. Ateach period, a senior can be in one the following five states depending onwhether he/she is alive and his/her nursing home and private LTC insur-ance coverage status. Let M represents the demise status, N the status ofnursing home and D the status of insurance coverage. Specially, M = 1means the state of demise, N = 1 means the senior is under nursing homecare, and D = 1 indicates that the senior is covered by some private LTCIpolicy. There are only five statuses because the demise status M = 1 is anabsorbing state: once an individual is dead, the values of the other statevariables don?t matter. The other four states are thus a combination of theN and D conditional on M = 0, e.g., N = 1 and D = 0 indicates the senioris staying at a nursing home facility while not covered by any private insur-ance policy. Other state variables are the age of the senior at current perioda, the liquid asset of the individual w, the housing asset of the individual h,the annual income y, and subjective risks ?. However, the subjective risks ?will not be implicitly appeared in our dynamic equations. Rather they areexplicitly embedded in expectation functions. For more discussion, readersare referred to section 4.4.3.At any age a < A, the senior will die next period with some probability1084.4. Model: Specification, Identification and Estimationand after reaching age A, the mortality risk is assured. Further, individualsobtain utility only from bequest they left if they die during the currentperiod. Thus, the value function is given byV (a,w, h,D,N,M = 1) = bi(w + h) (4.1)the welfare from unconsumed wealth after death can be a result of strategicbequest motive (Bernheim, Shleifer and Summers [1985]) or altruism (Tomes[1981]). Though we are not interested in the structural interpretation ofbequests per se, we will come back to discuss the relationship between expost expected value of bequest motive and children. Note specially, theutility function is idiosyncratically specified by the subscript i. When itdoes not cause confusion, however, we omit all the subscripts i standing forthe discussed agent.Equation 4.1 also implies that the model distinguishes two different as-sets w and h. We do so because of the eligibility and recovery policies ofMedicaid as discussed above. Introducing two different assets brings severalissues. First, we need to specify how an asset can transform to anotherasset. Under the Medicaid policy, asset transforming can be attractive de-pending on the distribution of assets. To exclude complexity, we assumethat individuals can not transform assets from one to another. More specif-ically, we allow the dynamics of liquid assets to depend on the consumptionchoice and make the housing asset fixed up to a real interest rate r. Thisassumption, plus an additivity assumption of utility, thus normalizes theutility from real estate assets to be zero. Second, the initial distribution ofthe two assets is exogenous in the model. This is equivalent to saying thatthe manipulation of the asset portfolio has already been done before retire-ment age. Admittedly, the exogeneity of initial distribution of two assets isa strong assumption since the distribution would depend on unobservableheterogeneity in risk aversion and bequest motive. A more satisfying modelshould explicitly specify a function between preferences and the distributionby either speculation or assumption. We did not choose to do so becauseof extra computation burden. However, we did examine how ex post risk1094.4. Model: Specification, Identification and Estimationaversion and bequest motive is related to the initial asset distribution andonly found some mild correlation 5.The uncertainties in the model come from the uncertainty about nursinghome care 6 and mortality risk. More specifically, the duration of nursinghome stays is assumed only to last one period and be identical for eachindividual 7. However, we allow the expense of nursing home care B tobe idiosyncratic but known by each individual ex ante. Restricting theduration of nursing home simplifies unnecessary analysis complication andalleviate computational burden. Otherwise, we need to further specify thetransition matrix of consecutive nursing home events (which can be age andheath dependent). In addition, the restriction does not adversely affect theestimate of the interesting preference distribution.Last, the model implicitly assumes all expectations in the model are thoseself reported by seniors, rather than those inferred using ex post realizationsas the conventional rational expectations method. Two expectations, thoseof mortality and of entering a nursing home facility, are the major uncertain-ties in the model. By adopting self-reported subjective probabilities, we donot need to implicitly specify age or health status variables when it does notcause confusions since those variables should be integrated into subjectiveexpectation by seniors. For more discussion, readers are once again referredto section 4.4.3.The timing of the model is the following. At the beginning of eachperiod, the mortality uncertainty is resolved and thus known. If a senior is5More concretely, we define the distribution as the ratio of liquid asset to total asset.In an OLS regression of the ratio on ex post risk aversion and bequest motive, though theinteresting coefficients are statistically significant, the R2 value is merely 1.2%. However,keeping in mind that our estimates of risk aversion and bequest motive are based onthe initial asset distribution, the statistic significance does not necessary invalidate ourexogenous assumption.6It implies that nursing home status is determined by exogenous factors like the dif-ficulties of activities of daily living (ADLs) and other supply factors like availability ofnursing home care beds. Though to what extent seniors can choose to enter in to a nursinghome ex post is an empirical question, it is reasonable to claim that entering into a nursinghome is a random event ex ante since the seniors have to figure out the risk while makingthe policy purchasing decision.7The average duration in a nursing home is about 2.04 years (Society of Actuaries[2007]), which corresponds to one period in our data set.1104.4. Model: Specification, Identification and Estimationalive, she/he has to decide the consumption level and whether to pay theinsurance premium or not. Each paid premium covers the this period only.And the premium is only determined by the initially purchasing age.An individual obtains utility from consumption while alive. However,the individual does not evaluate the hospital care while in a nursing homefacility. In other words, the service offered by a nursing home facility doesnot enter into the utility function. It is not clear how this affects our pref-erence estimate. On one hand, if individuals anticipated the welfare from anursing home facility and took it into account in decision making, ignoringthe utility might overstate the mean value of risk aversion but underesti-mate bequest motive parameter. Conversely, treating nursing home care asneutrals (not goods nor bads) ignores the ability of workers to offset longterm care shocks by adjusting their consumption. This leads us to incon-sistently estimate the consumption risk facing uninsured individuals, andthus the mean value of risk aversion and the bequest motive. On the otherhand, maintaining this assumption allows us to shut down the moral haz-ard channel in our model: obtaining utility form nursing home care wouldpresent some incentive to a senior to choose to enter into a nursing home.Therefore, nursing home risk is only treated as a financial uncertainty. Morespecifically, for D = 0 or D = 1, the value function of an a years old seniorwith liquid asset w and housing asset h who is in a nursing home facility isgiven byV (a,w, h,D,N = 1,M = 0) = maxc[u(c)+ (4.2)? ? [Prob(M ? = 0|N = 1,M = 0) ?W (a?, w?, h?, D? = 0, N ? = 0,M ? = 0)+Prob(M ? = 1|N = 1,M = 0) ? V (a?, w?, h?, D?, N ?,M ? = 1)]]where ? is the per-period discount rate, the superscript prime ? designatesnext period, Prob(M ? = 0|N = 1,M = 0) and Prob(M ? = 1|N = 1,M = 0)are the probability of death and living at the beginning of next period con-ditional on current nursing home status N = 1. In the above two mortalityprobabilities, age is intendedly omitted since in our empirical analysis these1114.4. Model: Specification, Identification and Estimationprobabilities are extracted from self-reported subjective mortality probabil-ities (see subsection 4.4.3).And the value function after experiencing and leaving a nursing home inthe past is given byW (a,w, h,D,N = 0,M = 0) = (4.3)maxc{u(c)+?EW (a?, w?, h?, D? = 0, N ? = 0,M ?)}which is subject to a transformation condition of w? and h?. The constrainton N ? = 0 reflects the assumption that a senior can not enter in to a nursinghome again after being dispatched. SinceN ? = 0 holds for the future periods,there is no need to pay the insurance premium to get covered, D? = 0. Thisequation explicitly assumes all seniors at most have one chance to enter anursing home facility. In other words, once they left a nursing home (if theydid so before death), they either die or stay home next period. In practice, atypical senior usually visits nursing homes several times in her/his life time.However, multiple visits are irrelevant in our model since for a senior tomake the purchasing decision what matters is the expected expense of longterm care. Notice also we use value function W () to denote dynamics afterexperiencing a nursing home here.Equation 4.2 reflects the assumption that the nursing home care bringsno welfare and the only utility sources are either from current and futureconsumption, or the bequest. In this equation, it implicitly allows a seniorto choose an optimal consumption level to maximize utility level even whenthe senior is in a nursing home. In practice, seniors in a nursing homeoften sadly found they were aboard a boat preventing such optimization.However, completely removing such optimization in our analysis demandsan alternative constrained optimization, which is even harder to justify.We have not yet determined the dynamics of w?and h?, which depend onboth the insurance coverage and Medicaid policy. Consider the case whereD = 1. Since a private insurance policy covers all the expenses of care, we1124.4. Model: Specification, Identification and Estimationhavew? = (w + y ? c) ? (1 + r) (4.4)h? = h ? (1 + r) (4.5)where r is the per-period real interest rate. In this value function, we ignorethe co-payment and deductable the individual needs to pay.Consider the D = 0 case now. First, if the senior dies at the beginningof next period M ? = 1, the Medicaid recovery and spending down policiesindicatew? + h? = max{0, w + h+ y ? c?B} ? (1 + r) (4.6)where B is the known, idiosyncratic nursing home care expense. Second, ifthe senior survived after the nursing home care, that is M ? = 0, then thedynamics are given byw? = max{2000, w + y ? c?B} ? (1 + r) (4.7)h? = h ? (1 + r) (4.8)The w? transformation reflects the Medicaid spending down policy in termsof the liquid asset and 2000 is the Medicaid allowance for each single seniorthat can be exempted from the spending down policy. The dynamics of w??and h?? depends on the M ??. If the senior is still alive at age a??, the dynamicsof w?? and h?? are identical as the dynamics of w?and h? in equation 4.7 and4.8. However, if M ?? = 1, the estate recovery policy is executed and thedynamics becomeh?? + w?? = max{0, (w? + h?) ? (1 + r) +min{0, w + y ?B}} (4.9)where the inner min calculates the balance Medicaid paid on behalf of therecipient after the recipient spends down all the liquid asset. And w+y?B >0, that is the individual is rich enough to pay the nursing home care billout-of-pocket, then Medicaid paid nothing and can not execute the recoverypolicy. The outer max is obvious. Strictly speaking, the states indicating1134.4. Model: Specification, Identification and Estimationwhether a senior ever in a nursing home should also be a state variable sinceit affects the later state space of N after a senior leaves the care facility.If a senior is not in a nursing home, which is N = 0, at the beginning ofthe period, the senior decides whether or not to buy a private LTCI policy inadditional to the usual optimal consumption. If the senior is not covered byany insurance policy D = 0, he/she can choose to buy a new policy and paythe premium determined by current age a. If not covered by any insurancepolicy D = 1, he can choose either to let the policy lapse and not pay anypremium or continue paying the same premium as last period p(a ? 2). Inthis period, the senior faces a trade-off between buying an insurance policythis period to prevent the financial risk of next period and not buying apolicy to maintain a higher expendable liquid asset. Moreover, for D = 0 orD = 1, the senior?s value function is given byV (a,w, h,D,N = 0,M = 0) = (4.10)max{V b(a,w, h,D,N = 0,M = 0),V n(a,w, h,D,N = 0,M = 0)}where V b() and V n() stand for the value of current states if the senior decidesto buy a policy and not to buy a policy respectively, and are given byV b(a,w, h,D,N = 0,M = 0) = (4.11)maxc{ui(c)+?EV (a?, w?, h?, D? = 1, N ?,M ?)}subject tow? = (w + y ? c? p(a)) ? 0 if D = 0w? = (w + y ? c? p(a? 2) ? 0 if D = 1andV n(a,w, h,D,N = 0,M = 0) = (4.12)maxc{ui(c)+?EV (a?, w?, h?, D? = 0, N ?,M ?)}1144.4. Model: Specification, Identification and Estimationsubjective tow? = (w + y ? c)(1 + r) ? 0where a ? 2 is the age at last period 8, and p(a) is the premium the seniorpays. We maintain the standard assumption that seniors can not borrowagainst future income with the presence of mortality uncertainty. The ex-pectation E is integrated over the distribution of the states of next periodnursing home, N ?. If the senior decides to buy a policy, then next period thesenior is covered by an insurance policy D? = 1. Otherwise, D? = 0. Theterminal condition is given by V (A,w, h,D,N,M) = bi(w+h) since demiseis a sure thing after age A.Since government transfers are generally available to bridge the gap be-tween an individual?s expendable income and the minimum consumptionfloor, the individual?s consumption can not be less than some c. Specially,we assume that the consumption level c automatically jumps to c once theoptimal level is below the floor.Finally, to recover the joint risk aversion and bequest motive distribution,we allow heterogeneity in both preferences. Concretely, the consumptionutility function is assumed to be a constant relative risk averse (CRRA)functionui(c) =c1??i ? 11? ?i(4.13)and the bequest utility function is assumed to bebi(w) = ?i(c+ w)1??i ? 11? ?i(4.14)where c and w are consumption and bequest, ?i and ?i are the risk aversionand bequest motives parameters. Recovering the distribution of the two8Notice that one period corresponds to two years in the model. In practice, the pre-mium paid, conditional on the coverage history, depends on the age when the elderlyinitially bought the policy. The insurance companies frequently adjust the cohort pre-miums, though they can not charge a higher premium only because of aging. Note thatp(a?2) can be either the premium paid by a senior (which is determined by the age whenthe policy was initially bought) or the premium charged it it is sold at age a? 2.1154.4. Model: Specification, Identification and Estimationparameters is the main task of the paper. Notice c is the minimum con-sumption level below which no bequest will be left and thus determines theextent to which bequests are luxury goods. In a static model with the abovepreferences, the optimal bequest level is a fraction of the after minimumconsumption w ? c. Similar bequest preferences can be found in De Nardiet al. [2010] and Lockwood [2011] etc..In this model, both risk aversion and the bequest motive have the nicefeature of monotonicity 9: a larger value of ?i implies a stronger risk aversionand thus a stronger desire to purchase an insurance policy while a greatervalue of ?i implies a stronger bequest motive and thus a stronger desireto buy an insurance policy. Intuitively, all else equal, we expect that asenior with higher bequest motive would be more likely to buy an insurancepolicy. Since the risk aversion parameters appear in both consumption andbequest utility function, a higher level of risk tolerance makes insurancemore desirable. In other words, observing the insurance decision helps usto infer both the value of the risk aversion parameter ?i and the bequestmotive ?i.Several features of the model need to be mentioned. First, we onlymodel the uncertainties on demise and nursing home status of next period.All uncertainties on income y and the policy premium p(a) are omittedfrom the model. It is a natural choice since we are mainly interested inthe discrete choice of buying insurance, not the volatility of consumption.Besides, income y of seniors consists of social security and pensions etc.,which do not depend on employment status. Second, according to our model,switch of insurance coverage is mainly driven by the change in the perceivedrisks. In reality, as showed recently by Knotzka and Luo [2010], most ofthe policy lapses are driven by deterioration of financial status. To capturefinancial uncertainty, we could alter the model with a random income shocky, instead of fixing it. We choose not to do so because of the attempt tokeep the model simple. Third, this is a partial equilibrium model where the9The monotonicity feature actually is quiet broad. In includes many preferences thatfollow the specified utility functions. See Lockwood [2011] for a discussion on the bequestutility function.1164.4. Model: Specification, Identification and Estimationpricing strategy p(a) and the Medicaid policies are determined outside themodel. A fully equilibrium model would be very attractive for answeringsome general equilibrium effects like how Medicaid eligibility rule changescan affect the welfare of seniors from different social groups. Last, otherdecisions that are observable in the data set and help to infer preferenceinformation, e.g., life annuity purchases and choices in the Medigap market,are not integrated in our model. Doing so provides more vehicles to inferpreference, but the estimated preference parameters will be entangled byvarious decisions.The observation of seniors? choices plus the rationality assumption thatthe choices are made optimally according to the specified model providesinformation to retrieve underlying preferences. Intuitively, everything elseequal, buying an insurance policy is optimal if and only if a senior is riskaverse enough; and it is also true if and only if he/she has a strong enoughbequest motive.4.4.2 Preference IdentificationThere are two types of parameters in our model: one type about preferences? and ? and another type about beliefs: nursing home expenditure B andsubjective probability assessment of mortality and long term care risk. Toestimate the model, we separately estimate these parameters following Rustand Phelan [1997] and Gourinchas and Parker [2002] etc.. Unlike others,beliefs in our data are partially observable thus our focus is on the preferenceparameters. This subsection discusses the identification of preferences thatis followed by the discussion on beliefs.To understand how to achieve identification, we first consider the casewhere seniors making LTCI purchase decisions strictly follow a monotonicrule, that is d = 1 if and only if V b > V n. Without any utility valueshocks, buying is optimal if and only if the risk aversion preference is highenough, and for these currently covered by some private policies, not buyingis optimal if and only if the risk aversion preference is low enough, everythingelse equal. Similarly, for these not covered by any private policies, buying1174.4. Model: Specification, Identification and Estimationis optimal if and only if the bequest motive is strong enough, and for thesecurrently covered by some private policies, not buying is optimal if and onlyif the bequest motive is low enough, everything else equal.To better understand the intuition, we draw a figure which shows thearea where it is optimal for a senior to buy an insurance policy. In figure4.1, the horizontal line represents bequest motive while the vertical linerepresents risk aversion. First, the monotonicity of the decision regardingrisk aversion can be intuitively understood that there is a cutoff value of??i (?i) such that, holding the bequest motive constant, the senior will choosedi = 1 if and only if the value of ?i ? ??i (?i). Second, a similar conclusion canbe made for the bequest motive parameter. Combining the above resultsgives an indifference curve, which is the locus of all the ?i and ?i suchthat, conditional on the current status D, buying an insurance policy bringsthe senior the same state value as choosing not to do so. Further, underany value combination of (?i, ?i) above the indifference curve, the seniorwill choose to buy a policy di = 1 and under any combination below theindifference curve, the senior will choose not to buy di = 0. Thus, the areabelow the indifference curve and the two axes gives the probability that thesenior will choose not to buy a policy at the cumulative distribution functionF (?, ?). Since we already expressed the probability of not buying a policy asa function of the joint distribution of risk aversion and bequest motive, wecan recover the joint distribution function by maximizing likelihood basedon the decision choices.To formally prove the identification idea, we introduce some notationfirst. The observed data on each individual i who is alive Mi = 0 andnot in a nursing home facility Ni = 0 are (Di, di, wi, hi, ?i, xi), where Diis observed insurance coverage states, di is the insurance policy purchasingdecision which is 1 if the individual decides to buy a policy and 0 otherwise,wi and hi are the liquid and housing asset respectively, ?i is the subjectiverisk reported by the senior and xi is a vector of observed characteristics.Identification is achieved if and only if the joint distribution function F (?, ?)is uniquely recovered given enough observations.The identification requires the following three assumptions:1184.4. Model: Specification, Identification and EstimationFigure 4.1: Optimal Decision Without Utility ShocksFigure 4.2: Optimal Decision With Utility Shocks1194.4. Model: Specification, Identification and EstimationAssumption. The purchasing decision di is optimally chosen by individualsdescribed in the model.Assumption. Preference distribution log(?i, ?i) is i.i.d. drawn from an un-known normal distribution N(?,?),where ?i = (?? , ??) and ? =[?2? ?????? ?2?]is the variance and covariance matrix of (?, ?).Assumption. The premium p(a) and nursing home care expense B satisfythe condition: for all the a, N and w,V ?2(a,w + (1 + r) ? p(a? 2), h,D = 1, N,M = 0) (4.15)? V ?2(a,w, h,D = 0, N,M = 0)where V ?2 is the first order derivative of V w.r.t. the second term w.The first two assumptions are standard. Without assumption 1, the datacan not reveal any useful information about preference under the modelsetup. Without the distribution assumption, we could not evaluate the like-lihood of the observed data. Note assumption 2 also implies ?i > 0, that isseniors are risk averse. The crucial assumption is the last one, which liter-ally states the benefit of getting covered currently by paying the premiumin the last period is big enough in the sense that the marginal value of liquidassets under coverage, even compensated by the premium paid p(a? 2)) isless than the marginal value of liquid assets if not covered by any insurance.In a very coarse understanding, the premium p(a) and the nursing homecare expense B under current Medicaid rules should make getting coveredattractive marginally. When a = 2, the condition holds trivially given b() isconcave. To fully appreciate assumption 3 is difficult without presenting thekey proposition for the identification. Thus, we present the key propositionand then move back to discuss the last assumption.Lemma. If the above three assumptions hold, then di = 1 if and only if(?i, ?i) is above the indifference curve (??, ??) , which is the locus suchthat V bi (a,w, h,D,N = 0,M = 0) = Vni (a,w, h,D,N = 0,M = 0) for1204.4. Model: Specification, Identification and Estimationall a ? A? 1.Proof. The difficulty of the proof lies in the fact that the value functiondoesn?t have a closed form solution10. To overcome it, we exploit the finitehorizon recursive structure of the model. We omit the state variables Mand N since both take the value zero. Consider the case when D = 0. LetV ? = V b(a,w, h,D = 0) = V n(a,w, h,D = 0) at some value ??. The FOCof V b gives u?(cb) = ?(1 + r)EV ?2(a?, w?, h?, D = 1, N ?) and FOC of V ngivesu?(cn) = ?(1 + r)EV ?2(a?, w?, h?, D = 0, N ?). Consider three cases. First,cb = cn, which can?t be true by assumption 3. Second, let cb < cn. Itimplies u?(cb) > u?(cn) and w?b > w?n ? p(a)(1 + r), again this is impossibleby assumption 3. So, the only case is cb > cn at ??. Now consider a verytiny increase in ??, say d?. Since it is very tiny, the increase only has firstorder effects on the value of u()and V (), and negligible second order effect oncband cn. Since u?? < 0 , a d? increase will drop the value of u(cb) andu(cn).And since u???c < 0, u(cb) drops less than u(cn). Since the value functionV () inherits the properties of the utility function u(), a similar result canbe found for the second term of the V b() and V n(). Combining these twogives that V b() > V n() after a tiny increase in ??. For values ? > ??, thesame conclusion can be made based on the same reasoning. A similar resultcan be found for D = 1. Finally, we have not yet shown the existence of??. However, this can be easily achieved by noticing the monotonicity of thevalue function, which is omitted.From the above proof, it is clear that assumption 3 is critical in buildingthe conclusion that cb > cn on the indifference curve. Notice that cb is theoptimal consumption level if a senior chooses to buy an insurance policy,while cn is the optimal consumption if the senior does not buy. In a staticworld, the optimal consumption of not buying should be larger than theoptimal consumption of buying since the latter has to pay an extra premium.In a dynamic world, it does not hold anymore because to make a buyer and10A strict proof is very difficult to retain therefore the provided proof is very heuristic.However, the property is validated during empirical analysis.1214.4. Model: Specification, Identification and Estimationa non-buyer indifferent on lifetime utility, the non-buyer has to save moreto compensate for the utility at all future uncertain life periods.The above proposition presents a monotonic decision rule that jumpsat some indifference curve (??, ??). However, it literally implies there areno unobservable utility shocks and decisions strictly follow a deterministicrule. In reality, a deterministic decision model fits data less successful thana model allowing utility value shocks. Following the work of Rust [1987], weassume the following decision ruled = 1(V b + b ? (V n + n)0) = 1(n ? b ? V b ? V n)where b and n are shocks to utility values V b and V n and follow type Iextreme value distribution with variance ?2 =pi26 . Thus,  =b?n?followsthe logistic distribution, which gives the probability of choosing d = 1,Prob(d = 1) =exp((V b ? V n)/?)1 + exp((V b ? V n)/?)Figure 4.2 shows the random decision rule where the probability of buyinga policy at each point is given by a logit model.4.4.3 Belief Estimate and Parameters CalibrationAccurately pinning down beliefs on risks of entering into nursing home facil-ity, mortality and long term care expense is necessary for identifying prefer-ence parameters. Generally, belief is a result of information retrieving andevaluating process with abundant unobservables. As discussed in chapter 3,this complicated process is usually simplified to an ex post calibration underrational expectations assumption, which essentially states that observed expost occurrence is good enough to infer ex ante expectations. This is usuallythe only practical choice since in most situations no subjective data with re-spect to individual beliefs is available. However, one feature of the HRS datais the extensive, detailed information on seniors? subjective probabilities onvarious uncertainties like long term care and life expectancy.Belief on nursing home expense B is not asked in the HRS data set.1224.4. Model: Specification, Identification and EstimationTo overcome the problem, we assume that belief is rational: the expectedexpense B is identical with the true expenses for individuals ever admittedinto a nursing home facility. Since the true expense is not known for indi-viduals never admitted into a nursing home facility, we impute the expenseB by adopting a two-step method.Why Subjective Risks The objective of the paper is to understandhow seniors make long term care insurance decisions by recovering risk andbequest preferences from observing insurance purchasing choices made byforward-looking rational seniors. As Manski [2004] argued the classical iden-tification strategy hinges on the rational expectations assumption, which as-sumes ex ante belief identical with risk inferred from the ex post realization(rational risk). However, the previous chapter clearly shows that seniorssystematically misunderstand some information and form a biased ex antesubjective expectation about future nursing home utilization. Spinnewijin[2009] found evidence in support of the biased perceptions of risk by show-ing empirically that unemployed workers overestimate how quickly they willfind work, but underestimate the return to search efforts. Snowberg andWolfers [2010] found evidence in favor of the view that mis-perceptions ofthe winning probability drive the favorite-long shot bias in racetrack gam-bling markets. The discrepancy between the ex ante subjective belief andrational risk presents a serious challenge to the traditional treatment.Besides the expectation bias, beliefs inferred from the rational expecta-tions assumptions does not capture private information an individual holdswhile forming his/her subjective beliefs. For example, it is very commonin our data set to observe two individuals with identical observable charac-teristics reporting different subjective risks about future nursing home needand mortality. Finkelstein and McGarry [2007] compared individuals? sub-jective risk of entering a nursing home with the risk assessed by insurancecompanies? risk table and found that individuals have some residual privateinformation regarding the risk. With the presence of expectation bias, itis conceptually inferior to calibrate the decision process without capturingprivate information.1234.4. Model: Specification, Identification and EstimationLast, even if ex ante subjective risk is free of biases, using ex post re-alization to assist identification is inappropriate because of moral hazard.If seniors under insurance coverage are more prone to enter into a nursinghome, belief inferred from ex post realization, no matter what expectationassumptions were made, is always inconsistent because the realization in-cludes the moral hazard information as well. More specifically, consider anexample in our context where a senior with rational expectations and moralhazard makes a decision to buy a LTC insurance policy. The realization ofthe random event is more likely to be observed because of moral hazard.Without taking it into account, the inferred ex ante risk will be higher thanand thus inconsistent with the true ex ante belief on which the senior?s de-cision is based 11. Since moral hazard is a rampant phenomena in variousinsurance markets, a traditional treatment of assuming its absence wouldnot obtain consistency.The major challenge to subjective risk is the validity of subjective data.However, chapter 3 has shown subjective risk actually predicts realizationpretty well, and evidence of good or better performance of self reportedsubjective probability in explaining observed choices has been confirmed inother research. For example, Juster [1966] found self reported purchasingprobability actually outperforms the verbally assessed buying intentions inexplaining the automobile purchase behavior. Hurd and McGarry [1995]found the HRS respondents self reported subjective probability of survivalaggregate to the actual population probability and predict the actual sur-vival. Hurd and McGarry [2002] presented evidence that the respondentsactually modify their survival probability in responding to new informationand the subjective survival probability do predict actual survival: those sur-vived in the survey panel reported about 50% greater probability than thosedied. In experiments, Nyarko and Schotter [2002] and Bellemare, Kro?ger andVan Soest [2008] found models using subjective probability data can gener-ate much better sample prediction than various ?rational? models where ex11I take a naive understanding about rational expectations here. A sophisticated ratio-nal individual might also take the moral hazard into account while forming the rationalexpectation.1244.4. Model: Specification, Identification and Estimationante probability is inferred from the ex post realizations.To summarize, we argue that subjective risks are a better choice thanrisks inferred from the rational expectations assumption in recovering prim-itive parameters and therefore both subjective nursing home risk and sub-jective mortality risk will be utilized in all our analyses.Transforming Subjective Risks Though self reported subjective risksutilize all private information, they are not designed for our analysis. Spe-cially, in the HRS data set, the subjective risk of entering a nursing homefacility is framed in a five year horizon while the mortality risk is in differ-ent horizons that depend on ages of seniors. In contrast, a period in ourmodel corresponds to two years. Therefore, subjective probabilities definedin other horizons need to be transformed to probabilities defined over a twoyear horizon. Typically, the subjective belief of nursing home care is elicitedby a formal specification of risk range from 0% to 100% and the followingsurvey question:What is the chance that you will move to a nursing home inthe next five years?which is immediately followed with a definition of nursing home12. Similarly,the subjective belief of mortality is asked by the following survey question:What is the percent chance that you will live to be 75 ormore?Obviously, the subjective belief on nursing home care is asked in a five yearhorizon while the mortality belief is surveyed in various horizons depend-ing on a senior?s age at the survey. In the following discussion, we showhow to transform the self-reported belief defined in a five-year horizon intoa two-year belief without losing the embedded private information. In ourview, there are two minimum standards that must be met for any reason-able transformation: first, the resulted probabilities should have the range12Chapter 3 gives details on subjective question on nursing home.1254.4. Model: Specification, Identification and Estimationfrom 0% to 100%. Second, for any two probabilities p1 and p2, the after-transformation probability of the former should be no less than that of thesecond one, tr(p1) > tr(p2), if p1 > p2. In other words, the transformationshould preserve the order of probabilities.Our starting point is to assume a hazard analysis process while an in-dividual assessing probabilities. Borrowing the jargon of survival analysis,an individual reports a risk by integrating hazard rate given by the mixedproportional hazard (MPH) model :hi(t) = aih0(t)exp(?0 +Xi?1)where hi(t) is the individual i?s hazard rate at time t, ai is idiosyncratichazard factor and represents the private information the individual holds,h0(t) is the common hazard rate for all individuals in the economy and Xiis the individual?s observable characteristics. To see how the MPH modelcan help in preserving the private information, notice the subjective risk isactually the cumulative probabilityFi(t), which is connected to the hazardrate though?ln(1? Fi(t)) =? t0hi(v)dv =? t0aih0(v)exp(?0 +Xi?1)dvThe primary objective here is to infer Fi(2) from Fi(5) since a period in ourmodel corresponds to two years. To recover Fi(2), we assume h0(t) = h0,which implies a constant baseline hazard factor. This seems, in our setup,an acceptable assumption. It states that the common hazard rate of thewhole society is independent of the duration. Under the constant commonhazard rate assumption, the risk is given by?ln(1? Fi(t)) = aiexp(?0 +Xi?1)h0tand Fi(2) is thus given byFi(2) = 1? (1? Fi(5))2/51264.4. Model: Specification, Identification and Estimationand similarly for other horizons.The proposed transforming method meets the standards discussed above.First, it preserves the fundamental property of probability: the transformedprobability has a range from zero to one. Second, it also preserves the privateinformation an individual holds in assessing the subjective probability in thesense of maintaining the order of probabilities. An other advantage is thatit exploits the key idea of survival analysis and does not impose strongassumptions.Last, we maintain the independent assumption of risk events. Allowingdependence between nursing home risk and mortality risk requires to specifyhow subjective beliefs interact. This is not an easy task without assuminghow seniors form and update subjective assessments.Though subjective risks of both nursing home care and mortality areobserved for each individual at current age, the beliefs for the rest of lifeare not available. In the simplest case, we could assume identical belief ofnursing home care or mortality at later life stages. Note, belief for the rest oflife is what individuals hold at current age for later life stages, not subjectivebelief held at later life stages. For nursing home risk, it seems acceptable toassume it does not vary with age. However, for mortality risk, it ultimatelysays individuals believe the risk of death is independent of age.Intuitively, there are two methods available to better recover the latersubjective mortality beliefs. The simplest one is to apply a nonparametricmatching process, where for each individual, according to his/her other at-tributes like gender, education and income etc., his/her subjective mortalitybeliefs at a later stage of the life a? are the mean value of the reported sub-jective mortality risks reported by those who are aged at a?. However, thismethod ignores life expectation improvement for younger cohorts and moreimportantly, ignores the private information of the individual included inhis/her current subjective risks. Another method, which is adopted in thischapter, to remedy this problem is to recover idiosyncratic mortality risks atlater life stages from the reported subjective risk of mortality at current ageby assuming an evenly increasing rate until the maximum age A at whichthe subjective probability should be one. Other methods of assuming an1274.4. Model: Specification, Identification and Estimationaccelerating increase or utilize other demographics characteristics are alsoconsidered in our computation.Imputation of the Belief of Expense B Since the expense B beliefis not asked in the HRS data set, we have to impute it somehow. Forthose ever admitted into nursing home care during our study period, theexpense B belief is assumed identical with true expense. For individualsnever admitted, the expense B belief is imputed using outside data sources13. During the period of writing the thesis, the only available data setaugmented with detailed health information and long term care expenditureis the National Long Term Care Survey (NLTCS) 2004. It is a longitudinalsurvey conducted by Duke University and designed to study changes inhealth and functional status of old Americans.In the first step, we use the NLTCS 2004 data to estimate the predictionequation for nursing home care expense. The equation is given byB = ?0 +X??1where X is a vector of personal observables, which consist of six healthmeasurements and four demographic variables. The vector X is observed inboth data sets so that we can use the estimated prediction equation fromthe above NLTCS data to impute the E(B?) for each individual in the HRSsample. At the second step, the expense B belief is assumed identical withthe estimated value E(B?). Equalizing the projected expense and the beliefseems restrictive because it implies that people form expectations followingthe OLS regression. However, the previous chapter has argued under ra-tional expectations, the coefficient of X should be identical in both beliefand true expense regressions. Hurd and McGarry [1995] noticed that themorality beliefs in HRS actually covary with other variables in the sameway actual outcomes vary with the variables in OLS regression. Further,the proposed method is only one convenient way for the purpose of identi-13Of course, we prefer to use the realized expense inside the HRS data set to projectthe expense. However, the admitted sample is too small to give a reasonable estimate.1284.4. Model: Specification, Identification and Estimationfication: any other methods that help to infer beliefs B would work as well.As a robustness check, we examine an alternative method of nonparamet-ric estimate of B for all seniors based on age, gender and education anddo not find significant effects on our following main results. Finally, forthe expected nursing home expenses at a later life stage, we assume eachindividual imposes a yearly 3% growth rate.Calibration of ?, r , c, A and p(a) We treat the parameters ? and r asobservables and calibrate their values. The discount factor ? is assumed tobe fixed at 0.98, and r is set at a value such that ??r = 1. The maximum ageA is set to be 100. Finally, we set c at 1000 dollars a month, approximatelythe Department of Human and Health Service poverty guideline level forone person household.Since premiums are only known for these who bought an LTCI, we haveto infer potential premiums for these who did not buy any private policies.To do this, we use a nonparametric matching method by noticing that pre-miums only depend on age and gender in each local market. We thereforematch the premium for each individual to the estimated mean premium foreach subgroup determined by the three factors.4.4.4 Likelihood FunctionThe likelihood function for each observation is given byli(di;?,?) = (4.16)1(di = 1) ? log[? ??? ??exp(V b?n(?, ?))1 + exp(V b?n(?, ?))?(?, ?)d?d?]+1(di = 0) ? log[? ??? ??11 + exp(V b?n(?, ?))?(?, ?)d?d?]where 1() is an index function, V b?n(?, ?) = Vb(?,?)?V n(?,?)?is the Bellmanvalue difference between buying and not buying decisions at each preferenceset (?, ?) divided by the scale factor ? and ?(?, ?) is the probability densityfunction of the bivariate normal distribution of log(?, ?). Thus, we have 61294.4. Model: Specification, Identification and Estimationparameters to estimate: scale factor ?, mean ?? and ??, variance ?? and??, and covariance ???. Ideally, we would prefer to have a closed form,continuous function of V b(?, ?) ? V n(?, ?) for the purpose of integration.However, it is not possibly achieved because of the unavailability of theclosed form solution to the Bellman value functions of V b and V n. Thus,we choose to calculate the likelihood by dividing the ? and ? space intomany small grids and calculating the cumulated probability of the smallgrid. Since the grid is very small, we thus can approximate the Bellmanvalue differences in the grid by the crossing middle point of the grid. In thisway, the likelihood is calculated asl(di;?,?) = (4.17)1(di = 1) ? log[n??j=1n??k=1exp(V b?n(?j , ?k))1 + exp(V b?n(?j , ?k))??(?j , ?k)]+1(di = 0) ? log[n??j=1n??k=111 + exp(V b?n(?j , ?k))??(?j , ?k)]where V b?n(?j , ?k) is the Bellman value difference at the grid (j, k) and??(?j , ?k) is the cumulative probability of the grid 14. When the grid sizegoes smaller, we can approximate the likelihood value closer but with morecomputational burden.To numerically calculate the likelihood, we first need to define an appro-priate range of ? and ?. The choice of bound limit of ? and ? is througha guess and verification process: pick some reasonable range and do themaximum likelihood estimation. If the estimated mean and standard errorshows the cumulative probability of the square defined by the range is atleast 0.99, then the range is chosen as the satisfied one; Otherwise, we in-crease the bound of the grid square. After defining the range of ? and ?,we need to select the appropriate step size to define the grid. At the initialattempt, we choose a random step-size, say 0.5 and do the estimation, if14Or precisely, ??(?j , ?k) = ?(?j+1, ?k+1)??(?j , ?k), the difference of CDF at the grid.1304.5. Resultsthe estimated probability at each grid is less then 0.01, then the selectedstep-size is the appropriate one; otherwise, we refine the step-size. Whilethis increases the computation, the selection can give us more precise mea-surement on the likelihood. With all these considerations, the range of [?, ?]and [?, ?] is set at [?5, 5] for log(?) and [?10, 10] for log(?) and the step sizeis 0.1. Thus, the grid is a 100 by 200 matrix for each individual. Notice,exp(?5) = 0.0067 and exp(5) = 148, which should include all reported riskaversion level in literature.4.5 Results4.5.1 The Distribution of ? and ?Table 4.3 reports the estimated distribution of the preferences and the vari-ance of utility shocks, ?. The estimated ?ln(?) is 0.2786, which correspondsto a mean value of ? 1.322. It is close to other estimated risk aversion usingdifferent methods and data sets. For example, Chetty [2006] reported anupper bound of ? < 2 on the curvature of utility over wealth by exploitingthe link between risk aversion and labour supply behaviour. The varianceof ln(?), ?ln(?), is only 0.0535. The small variance possibly reflects the sam-ple selection in our analysis: the seniors are singles with substantial assetsand similar educations. The bequest motive after taking the logarithm isestimated at ?3.3801, which is equivalent to ? = 0.0341. This value is farless than 1. Since in our setup, the bequest motive is similar to a scalefactor to the utility function, the small estimated ? implies that individualstreat unconsumed bequests as an inferior form of consumption. The smallmagnitude of the bequest motive seems to support the accidental bequesttheory rather than the altruistic model. If bequest is driven by altruism tooffspring, it seems reasonable to expect it to be close to 1: the person mak-ing the bequest prefers consumption left to his or her offspring at least thesame as his or her own consumption. Indeed, Hurd [1989] found a similarconclusion by estimating a life cycle model of consumption. He found themarginal utility of bequest is small, therefore he suspected most bequests1314.5. ResultsTable 4.3: Preference Distribution EstimatesEstimated Valued Standard Error?ln(?) 0.2786 0.0223?ln(?) -3.3801 0.8976?ln(?) 0.0535 0.0029?ln(?) 0.8314 0.5121? 0.0801 0.0373? 0.0116 0.0103Obs 696 696are accidental. The variance of ln(?) is 0.8314. Combined with the meanvalue ?ln(?), it implies a upper tail value of ln(?) at 95% is merely ?1.72,corresponding a value of ? at 0.18. It is still far less than 1. The magni-tude of ?ln(?) is more than 10 times of that of ?ln(?). Thus, we estimatesubstantial heterogeneity across individuals in their bequest motives. Thecorrelation ? is 0.0801. Seniors have stronger bequest motive are also tendto be more risk averse, however the correlation is very weak.4.5.2 Evaluating ex post PreferencesThe ex post Expected Risk Aversion ? and Bequest Motive ? Oneway to evaluate the model and estimated preference distribution is to calcu-late the ex post expected risk aversion and bequest motives for each individ-ual and compare them with other sources in the data. In the HRS data set,seniors were asked to do some hypothetical job games which can be used toelicit risk aversion. Further, information on children and subjective proba-bility to leave a bequest offers a good opportunity to evaluate the estimatedex post bequest motive.The ex post expected risk aversion is defined asE(?|d : ?,?) =n??j=1?jp(?j |d : ?,?)where p(?j |d : ?,?) is the conditional probability of ? fall within grid j1324.5. Resultsgiven values of d, ?, and ?, which is defined asp(?j |d : ?,?) =n??k=1p(?j |d, ?k : ?,?)p?(?k : ?,?)where p?(?k|?,?) is the marginal probability of ? within grid ?k. Noticep(?j |d, ?,?, ?k) is by Bayes rulep(?j |d, ?k : ?,?) =p(d|?j , ?k : ?,?)p(?j , ?k : ?,?)p(d, ?k : ?,?)Notice p(d, ?k : ?,?) is given byp(d, ?k : ?,?) = p(d|?k : ?,?)p?(?k : ?,?)and p(d|?k : ?,?) can be expressed asp(d|?k : ?,?) =n??j=1p(d|?j , ?k : ?,?)p?(?j : ?,?)where p?(?j : ?,?) is the marginal probability of ? within grid ?j . Thus,the ex post expected ? isE(?|d : ?,?) =n??j=1?jn??k=1p(d|?j , ?k : ?,?)p(?j , ?k : ?,?)?n?j=1 p(d|?j , ?k : ?,?)p?(?j : ?,?)The ex post expected bequest motive is defined similarly.Hypothetical Job Game and ex post Risk Aversion ? In the HRSdata set, seniors were also asked to make a choice between a certain joband a risk job under hypothetical scenarios. The hypothetical game wasintroduced by the following words:Now I have another kind of question. Suppose that you arethe only income earner in the family. Your doctor recommendsthat you move because of allergies, and you have to choose be-tween two possible jobs. The first would guarantee your current1334.5. Resultstotal family income for life. The second is possibly better pay-ing, but the income is also less certain. There is a 50-50 chancethe second job would double your total lifetime income and a50-50 chance that it would cut it by a third. Which job wouldyou take ? the first job or the second job?If the respondent accepts the risky job, she/he is proposed to choose between:Suppose the chances were 50-50 that the second job woulddouble your lifetime income, and 50-50 that it would cut it inhalf. Would you take the first job or the second job?If the choice is still the risky job, the respondent is asked about the riskiestjob:Suppose the chances were 50-50 that the second job woulddouble your lifetime income, and 50-50 that it would cut it byseventy-five percent. Would you take the first job or the secondjob?Similarly, if the respondent?s choice to the initial question is the certainjob, she/he is asked a less risky alternative:Suppose the chances were 50-50 that the second job woulddouble your lifetime income, and 50-50 that it would cut it bytwenty percent. Would you take the first job or the second job?and the least risky alternative:Suppose the chances were 50-50 that the second job woulddouble your lifetime income, and 50-50 that it would cut it byten percent. Would you take the first job or the second job?Based on the answers to the job game questions, we can calculate theex post risk aversion grouped by the answers and test if seniors choosingless risky options have higher ex post risk aversion. Since very few subjectschoose other options but the least risky alternative (277 among 696), we1344.5. Resultsgroup all subjects choose the three riskier options together. Among thesechoosing the least risky option, the average risk aversion ? is 1.23 with astandard error of 0.49. The average risk aversion among those choosingriskier options is 1.19 with a standard error of 0.45. Our estimates pass thetest that those choosing the least risky option have a higher average riskaversion.However, our ex post risk aversion is much smaller than that impliedin these hypothetical games. By assuming a CRRA utility function overwealth u(w) = w1??i?11??i, it is possible to recover the risk aversion rangefor each respondent. For example, for a respondent rejecting a one-thirddownside job but accepting a one-fifth downside job, the upper and lowerbound the respondent?s risk aversion are0.5 ?21?? ? 11? ?+ 0.5 ?(2/3)1?? ? 11? ?=11?? ? 11? ?=? ? = 20.5 ?21?? ? 11? ?+ 0.5 ?(4/5)1?? ? 11? ?=11?? ? 11? ?=? ? = 3.76Since most the of respondents choose the least risky option, we thusexpect an average risk aversion greater than 3.76, which is much biggerthan the estimated mean of the risk aversion distribution. In a recent pa-per, Einav and Finkelstein et al. examined individual?s choices over severalemployer-provided insurance coverage options and one 401(k) investment.Essentially, they found individuals? choices in one of the insurance domainsare more predictive of the other insurance policies than of the 401(k) in-vestment decision. Our results, however, seem to support that risk aversionestimated from different domains actually has the same order but with dif-ferent magnitudes.Children, Subjective Bequest-Leaving Probability and ex post Be-quest Motive ? It is popularly believed that bequest motive should cor-relate with the presence of children. Under the altruistic theory of bequests,it is natural to predict this correlation. On the other hand, if the driving fac-tor of bequests is for consumption smoothing and bequest motive is purely1354.5. Resultsaccidental, it is not clear why the presence of children should be a goodpredictor of bequest motive. Empirically, Hurd [1987] analyzed the bequestmotive by using a ten year panel and found no wealth changes differencesbetween households with living children and without living children. Simi-larly, Jurges [2001] found that having children has no significant impact onhouseholds? wealth trajectories in German. We have a similar finding here.By grouping the seniors over the presence of living children, we ended with89 seniors without any children and 586 seniors with at least one children.The average bequest motive for these with at least one child is 0.052(0.049),which is very close to the 0.051(0.050) of these without any children 15.This finding arises an interesting question that is why people appre-ciate money left after death if there is no offspring to take the bequest16. Of course, a reasonable justification would be the presence of nieces ornephews that would potentially substitute offspring. More broadly, charitydonations, psychological factors and habitual preference would also be thepotential reasons. To further explore whether the presence of offspring havesubstantial effect on estimated bequest motive, we reestimate the preferenceparameters based on the presence of children. The estimated bequest motive?ln(?) is ?3.8543(1.2003) for those without any children vs. ?3.1003(0.9327)for those with at least one child, and the estimated risk aversion ?ln(?) is0.2865(0.0402) for those without any children vs. 0.2673(0.0314) for thosewith at least one child. Though the closeness of risk aversion is of expected,the independence of bequest motive and children is not surprising: it furtherconfirms the previous accidental bequest motive conclusion.We last test if the ex post bequest motive can be predictive of variousself reported bequest leaving probabilities. In the HRS data set, seniorswere asked to report the probability of leaving any bequest, the probability15Another related concern is that these seniors with children living close by can have analternative to home care given by children. A similar analysis shows that the estimatedbequest motive for these with at least one child living close by is 0.056(0.054), which isvery close to 0.051(0.057), the distribution for these without any children living close by.16Notice the specification of the bequest utility function 4.14 does not depend on thepresence of children and the bequest motive is defined as a scale parameter to the usualCRRA utility function.1364.5. ResultsTable 4.4: How Bequest Motive Predicts Subjective Bequest Probability?(1) (2) (3) (4) (5) (6) (7) (8) (9)Bequest Motive ? 34.96*** 38.46*** 23.77** 53.53*** 56.60*** 33.46* 105.68*** 113.93*** 63.63**(10.81) (11.71) (11.32) (17.33) (17.97) (18.35) (29.23) (37.50) (28.19)Controls:Age and Gender YES YES YES YES YES YESAsset, Income and Health YES YES YESR2 0.01 0.02 0.06 0.01 0.02 0.07 0.02 0.04 0.25Observations 418 418 418 680 680 680 671 671 671Notes: Standard errors are in parentheses. Column 1 ? 3 corresponds to leave any bequest, 4 ? 6 to leave bequestat least $10K, 7 ? 9 to leave bequest at least $100K. Dependent variable is subjective bequest probability. Bequestmotive is the ex post estimated value. A single asterisk denotes significance at the 10% level, double for 5%, and triplefor 1%.of leaving bequest greater than $10k and the probability of leaving bequestgreater than $100k. Though the mechanism and underlying factors influ-encing subjective bequest leaving probability are unknown, the subjectiveprobability of leaving a bequest should be a function of bequest motive,household assets and income etc.. Table 4.4 presents the regression resultsof various bequest leaving probabilities on ex post bequest motive. For ex-ample, column (1) of the table lists a coefficient of 34.96, which means anincrease in posterior expected bequest motive by 1 unit corresponds to anincrease of the probability of leaving any bequest by 34.96 percent. Theexpected posterior bequest motive is 0.051 with the same size standard de-viation. It implies a senior with one standard deviation increase in posteriorexpected bequest motive generally reports a higher probability of leavingany bequest by about 1.75%. Similar reading can applied to other columns.Obviously, in all of the regressions, the ex post bequest motive is a significantpredictor of various bequest leaving probabilities.4.5.3 Understanding the PuzzleOne motivation of this chapter is to understand the long term care insurancemarket puzzle where a positive correlation between LTC insurance coverage1374.5. Resultsand LTC incidence rate is not supported by data. One explanation is thereexists a second dimension of private information that negatively correlateswith risk status and therefore offsets the positive correlation. And our es-timate shows a significant heterogeneity of the bequest motive other thanrisk aversion among seniors.To formally compare the two factors in explaining the absence of posi-tive correlation, we employ a general approach proposed by Chiappori andSalanie [2000]. This approach estimates a bivariate probit of insurance cov-erage and risk incidence conditional on all variables X observed by insurers.Let y and z be dummy variables of insurance coverage and risk occurrencerespectively and ? and ? be a pair of jointly normal distributed errors, then???y = 1(X? + ? > 0)z = 1(X? + ? > 0)where 1(.) is an index function. Conditional on X observed by insurers,the classical asymmetric information theory predicts a positive correlationbetween ? and ? because of either adverse selection or moral hazard. Itcan be shown the estimated coefficient between ? and ? follows a ?2 dis-tribution with degree 1. The empirical lack of positive correlation leads tothe suggestion that private information on preferences, in addition to pri-vate information on risk status, is another major driving factor in LTCIshopping decisions. If so, a positive correlation should be observed aftercontrolling for the factor in the insurance shopping equation y (whether thepreference should be controlled in the risk incidence equation is irrelevantsince it should not affect risk incidence).Following Finkelstein and McGarry [2006], control variable X includesobservable individual health and demographic status and risk classificationassigned by insurance companies. Table 4.5 presents the estimated coeffi-cient and Wald test value of the coefficient is zero. The first column confirmsthe lack of positive correlation in the LTC insurance market if no preferenceis controlled. In the second column, we further control for the ex post riskaversion. The coefficient is still small and the ?2 value is too small to reject1384.5. ResultsTable 4.5: Positive Correlation Test(1) (2) (3) (4)Correlation between ? and ? .071 .058 .513 .460Wald Test of the correlation=0 (?2(1)) .374 .233 3.852** 3.484*Controls:Insurer Observables YES YES YES YESRisk Aversion YES YESBequest Motive YES YESObservations 660 579 660 579Wald ?2 value 3762.13 3490.11 7018.31 5583.58Notes: This table lists the estimate results from biprobit regressions. The dependentvariables are LTCI coverage and LTC risk incidence. Insurer observables include 21 de-mographic and healthy conditions. A single asterisk denotes significance at the 10% level,double for 5%, and triple for 1%.the zero hypothesis. The third column controls for the insurer observablesplus an ex post bequest motive. The coefficient jumps to 0.513 with a ?2indicating a significance level of 5%. The last column further controls forrisk aversion and the last result maintains. Consistent with our previousfinding, the results indicate that the bequest motive actually has a moresubstantial impact on LTCI shopping decisions.This finding clearly shows the heterogeneity on bequest motive is an im-portant factor in formulating seniors? insurance shopping decisions. Com-bined with the small mean value of the bequest motive, we suspect the de-cisions are mainly driven by a strong desire to smooth consumption ratherthan a altruistic one to leave offspring assets. The heterogeneity on riskaversion is less significant quantitatively and less important in explainingthe puzzle.4.5.4 Counterfactual Policy AnalysisIn spite of high long term care cost, only about 10 percent of the seniorsare covered by private insurance policies. Even among the highly selectedsample in our analysis, only 17% bought any policies. This thin market has1394.5. Resultsbeen extensively discussed in literature. Pauly [1990] argued that a LTCIcoverage?s primary object is to protect bequest, but it is already excessivebecause of imperfect annuities. Further, the substitution of formal care tochildren care makes a policy also less favorable. Brown and Finkelstein[2008] examined the interaction of the public Medicaid program with theprivate market for long-term care insurance and found that Medicaid couldexplain the lack of private insurance purchases for about two-thirds of thewealth distribution even if there were no other factors limiting the size of themarket. Goda [2010] exploited variation in adoption and generosity of statetax subsidies for private long-term care insurance and found the average taxsubsidy raises coverage rates 28 percent.In this section, we do two counterfactual policy analyses to shed lighton the thin market size issue. First, we consider a policy that subsidizes in-surance premiums by taxing bequest. During the study period, the federalbequest tax policy in the United States only taxes estates valued greaterthan $675, 000 with many exemptions. Taxing all estates literally increasesthe price of bequests and makes consumption and LTCI less expensive. How-ever, it is possible for the income effect to dominate the substitution effect.The subsidy to LTC insurance premium makes purchasing coverage moreattractive. Second, we can alternatively subsidize the LTC cost instead ofthe premium by taxing bequests. The insurance coverage becomes less at-tractive if the cost of LTC is smaller. On the other hand, income effects willincrease purchasing power and thus the market size.To conduct the analysis, we recalculate the probability of buying an in-surance policy for each individual in order to get an average probability. Wedo this both under the cases without any policy interventions and these withthe proposed policy interventions. The policy effect is defined as the ratio ofthe difference between average probabilities to the probability without pol-icy interventions. Table 4.6 presents the policy effects under various policycombinations. As expected, the market size increases with the subsidy onLTCI premiums increase. Actually, the market size would increase 14.86%at the subsidy rate of 20% even with a tax rate of 5% on bequests. A taxon bequests harms the market size significantly: a tax rate moving from1404.5. ResultsTable 4.6: Policy Effects under Various ScenariosTax Rate on Bequest: 5% 10% 20%Subsidy Rate on LTCI Premium:5% -1.35% -1.35% -11.49%10% 4.73% 1.35% -7.43%20% 14.86% 12.84% 2.70%Subsidy Rate on LTC Cost:5% 12.84% 5.79% -2.70%10% 6.75% 2.02% -5.41%20% -18.92% -20.95% -23.65%Notes: Policy effect is the ratio of the probability difference between ascenario with proposed policy interventions and that without a policyintervention to the probability without a policy intervention.10% to 20% on bequests will make the policy effect shrink from 12.84% tomerely 2.70%. A Subsidy on LTC cost is more complicated because of theopposing direction of the income and substitution effects. When a subsidyrate is fixed at 5%, the policy effect of subsidizing LTC cost is bigger thansubsidizing the premium. At a low level of subsidy rate, the income effectsof the cost subsidy dominates the total effects of the premium subsidy. Withthe subsidy rate on cost increasing, the individuals find they are over payingfor the same policy, which leads to the market shrinking.The above analysis provides some insight regrading average counterfac-tual policy effects on population, however, it does not tell much about het-erogeneous effect of the counterfactual policies across different groups de-fined by ex post expected values of ? and ?. Here, we classify seniors intofour groups according to the ex post expected value of ? and ? using the themedian values of the ? and ? as threshold values. By this way, a senior inthe upper ? and upper ? group has the posterior values of ? and ? above 50percentile. Similarly for other three groups. Table 4.7 illustrates the policyeffects across four groups under various policy combinations. Rather thanlisting the probability difference ratio across four groups, this table reports1414.6. Conclusionthe within-group average probability both before and after policy changessince the average level of probabilities would be very different across fourgroups. At the second row, it first presents the original average proba-bility of purchasing LTCI policies across different groups. As expected, theUpper?Upper? group has the highest level of tendency to buy a policy. Also,value of ? has a greater effect on the probability than that of ?. The averageprobability after a policy change is listed in following panels. Comparingthe after policy average probability and the original probability across thegroups, it is clear that different policies have heterogeneous effects on dif-ferent groups. This is specially obvious when tax rate on bequest is 20%.With a high level tax on bequest, the incentive for the Upper? groups to buyinsurance is less strong since the motive to leave bequest is simply reducedproportionally with the tax rate. When tax rate on bequest is low at 5%,substitution effects naturally dominate income effects. This is weakly truefor the policy subsidizing LTCI premium, which is found to have a greatereffect on Lower? groups. This could be arose if lower ? values are somehowassociated with the seniors who did not buy LTCI because of high premiums.4.6 ConclusionUnderstanding how seniors make long term care insurance decisions interestsboth policy makers and academic researchers. While long term care expenseis the largest single health and financial risk facing seniors in the UnitedStates, the LTCI market is relatively small. Any public policy proposed toimprove the market size and social welfare etc. must build on a thoroughunderstanding of how insurance decisions are made. The lack of a positivecorrelation between insurance coverage and the ex post risk incidence ratechallenges the classical prediction in the LTCI market.This paper offers a new perspective to understand the LTCI market byconstructing a structural dynamic discrete choice model. The innovation ofthe paper is to estimate a joint distribution of two unobservable preferenceparameters, risk aversion and a bequest motive. Introducing the bequestmotive in the LTCI market is helpful since it has been documented exten-1424.6. ConclusionTable 4.7: LTCI Purchasing Probability across Different Groups Before and After Policy ChangesGroups: Upper?Upper? Lower?Upper? Upper?Lower? Lower?Lower?Original Probability 20.3 19.2 16.3 14.4Panel A: 5% Tax Rate on BequestSubsidy Rate on LTCI Premium5% 20.0 19.1 16.2 14.410% 21.0 20.2 17.0 15.520% 22.3 21.9 18.7 17.5Subsidy Rate on LTC Cost5% 22.5 21.3 18.6 16.510% 21.0 20.2 17.4 15.720% 17.3 15.9 12.8 12.1Panel B: 10% Tax Rate on BequestSubsidy Rate on LTCI Premium5% 20.0 19.1 16.1 14.410% 20.5 19.2 16.6 14.820% 22.3 21.4 18.6 16.7Subsidy Rate on LTC Cost5% 20.7 19.8 17.4 15.510% 20.5 19.6 16.8 15.020% 15.9 14.9 13.4 11.8Panel C: 20% Tax Rate on BequestSubsidy Rate on LTCI Premium5% 17.1 16.6 15.7 13.610% 18.9 18.0 15.2 13.420% 20.2 19.3 17.2 15.3Subsidy Rate on LTC Cost5% 19.2 18.3 16.4 14.310% 19.0 18.1 15.5 14.120% 14.8 14.2 13.3 11.7Notes: This table lists the average probability of purchasing LTCI across different groups after and before a policychange defined by a combination of a tax on bequest and a subsidy on LTCI premium or on LTC cost. A groupis defined by the posterior expected value of ? and ?. For example, Upper?Upper? group consists of those with? and ? at its higher 50 percentile. The average probability before any policy changes is listed at the second rownamed Original Probability. The average probability after a policy change is listed in Panel A, Panel B and PanelC. Therefore, 20 at column 1 and row 1 of Panel A means the average probability of purchasing LTCI is 20% forthe Upper?Upper? group after the policy combination of a 5% tax rate on bequest and a 5% subsidy on LTCIpremium.1434.6. Conclusionsively that a bequest motive plays an important role in explaining seniors?saving and consumption behavior.Our estimates indicate substantial heterogeneity in bequest motive. Theex post estimated bequest preference is quite successful in explaining theinsurance market puzzle: controlling for bequest motive generates the ex-pected positive correlation. Thus, we conclude bequest motive heterogeneityplays a substantial role in offsetting the positive correlation. The estimate ofrisk aversion is consistent with other research. Because our model capturesinstitutional details, it is capable of providing tools to evaluate potentialpolicy effects.Future development would be to estimate the model in a panel dataframework. This requires a careful restructuring of the model for the purposeof identification and integrating fixed effects. Since our paper only focuses ona small highly selected sample, caution should be applied when associatingthe results to the whole senior population.144Chapter 5ConclusionThis dissertation discusses three topics in applied economics.The first essay examines the causal effect of social capital on individ-ual income by exploiting the historically determined pattern of family namedistribution in Chinese villages. Family name distribution impacts socialcapital through historical inter-lineage rivalry and cooperation. The esti-mates show a strong first order effect on male villagers, which implies aone standard deviation increase in social capital is equivalent to two to fouryears of education. No effects on female villagers were found. The genderdifferentiation could be accounted for by occupation difference: male vil-lagers? income mainly comes from market exchange, while female villagers?income comes mainly from home production. Using a simple model, it isdemonstrated that a village?s social capital determines its trade scope andtherefore income of its residents.The second essay proposes a general method to identify subjective ex-pectation bias. The method exploits an implication of rational expectationsthat requires the identical weight of an independent variable in projectingboth objective and subjective probabilities. The empirical analysis showsthat female seniors do not correctly internalize age information while maleseniors fail at internalizing income information. Though cognitive abilityand risk aversion can partially explain the results, they are not the sourcesof the identified biases.The third essay explores how seniors make long term care insurance(LTCI) decisions by developing a dynamic structural discrete choice modelwhere a rational, risk averse, bequest motivated senior has to decide at eachperiod whether to buy an insurance policy or not 17. Using the Health17The rational model of this essay does not necessary conflict with the finding in the145Chapter 5. Conclusionand Retirement Survey data, this essay finds substantial heterogeneity inbequest motive that drives LTCI decisions. Specially, the idiosyncratic be-quest motive helps to explain why LTCI holders do not experience a higherincidence rate than non-holders.second essay. Expectation bias or rational expectation assumption are both merely onevehicle for the purpose of identification and expectation bias does not imply utility max-imizing behaviour. 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