NBER WORKING PAPER SERIES A SIMPLE TEST OF PRIVATE INFORMATION IN THE INSURANCE MARKETS WITH HETEROGENEOUS INSURANCE DEMAND

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1 NER WORKING PAPER SERIES A SIMPLE ES OF PRIVAE INFORMAION IN E INSURANCE MARKES WI EEROGENEOUS INSURANCE DEMAND Li Gan Feng uang Adalbert Mayer Working Paper NAIONAL UREAU OF ECONOMIC RESEARC 1050 Massahusetts Avenue Cambridge, MA January 2011 he views expressed herein are those of the authors and do not neessarily reflet the views of the National ureau of Eonomi Researh. NER working papers are irulated for disussion and omment purposes. hey have not been peerreviewed or been subjet to the review by the NER oard of Diretors that aompanies offiial NER publiations by Li Gan, Feng uang, and Adalbert Mayer. All rights reserved. Short setions of text, not to exeed two paragraphs, may be quoted without expliit permission provided that full redit, inluding notie, is given to the soure.

2 A Simple est of Private Information in the Insurane Markets with eterogeneous Insurane Demand Li Gan, Feng uang, and Adalbert Mayer NER Working Paper No January 2011 JEL No. D82,G22,I11 ASRAC A positive orrelation between insurane overage and ex post risk an be an indiator for private information in insurane markets. owever, this test fails if agents have heterogeneous risk attitudes. We propose a new test that onditions on unobserved types of individuals who differ in their risks preferenes. his makes it possible to detet asymmetri information without diret evidene of private information - even if agents have heterogeneous risk attitudes. We apply our tehnique to the market for long-term are insurane. Finkelstein and MGarry (2006) provide diret evidene for the existene of private information in this market. At the same time they fail to find a positive orrelation between insurane overage and ex post risk. Our method indiates the existene of private information, without using diret evidene of private information. Our methodology is appliable to other insurane markets and markets where proxies for private information are not available. Li Gan Department of Eonomis exas A&M University College Station, and NER gan@eonmail.tamu.edu Adalbert Mayer Department of Eonomis Washington College Chestertown, MD amayer2@washoll.edu Feng uang Researh Institute of Eonomis and Management Southwestern University of Finane and Eonomis Chengdu, China phenix.hf@gmail.om

3 1. Introdution Eonomi theory suggests that the presene of private or asymmetri information has important impliations on insurane markets. 2 Adverse seletion and moral hazard an lead to a sub-optimal provision of insurane and a derease in welfare. herefore, it is important to detet and to quantify the effets of asymmetri information in insurane markets. One indiator for the presene of asymmetri information is a positive orrelation between an individual s risk and the deisions to purhase insurane (after ontrolling for publi information). 3 his indiator does not require to diretly observe private information and it has been used to investigate a number of insurane markets. he empirial results, however, are mixed and differ by markets. For example, in a life insurane market, Cawley and Philipson (1999) onlude that the mortality rate of U.S. males who purhase life insurane is below that of the uninsured, even when ontrolling for many fators suh as inome that may be orrelated with life expetany. In an auto insurane market, Chiappori and Salanié (2000) find that aident rates for young Frenh drivers who hoose omprehensive automobile insurane are not statistially different from the rates of those opting for the legal minimum overage, after ontrolling for observable harateristis known to automobile insurers. In ontrast, Cohen (2005), using data from Israel, shows that new auto insurane ustomers hoosing a low dedutible tend to have more aidents, leading to higher total losses for the insurer. In an annuity insurane market, Finkelstein and Poterba (2004) observe systemati relationships between ex post mortality and annuity harateristis, suh as the timing of payments and the possibility of payments to the annuitant s estate, but they do not find evidene of substantive mortality differenes by annuity size. e (2009) draws a omplete different 2 Rothshild and Stiglitz.(1976) 3 Chiappori (2000) Chiappori and Salanié (2000) 2

4 onlusion with the existing literatures, by produing evidene for the presene of asymmetri information in the life insurane market. In partiular, she presents a signifiant and positive orrelation between the deision to purhase life insurane and subsequent mortality, onditional on risk lassifiation. One existing explanation for failure to detet the private information is the presene of heterogeneous preferenes for insurane. De Meza and Webb (2001) suggest that there may be advantageous seletion, whih means that more autious people are not only more inlined to purhase insurane but also more likely to put effort in preventing risk exposures. he presene of both adverse seletion and advantages seletion may reate insignifiant or even negative orrelations between an individual s risk exposure and the deision to purhase insurane even with private information. Finkelstein and MGarry (2006), short F&MG, introdue variables that measure autiousness and wealth quartiles. hese variables are expeted to be orrelated with insurane demand and risk exposures. hey find that these variables positively related to insurane demand and negatively related risk exposures. hey explain this result with the existene of multiple types of insurane ostumers. More autious and wealthier individuals are more likely to purhase long-term are insurane and less likely to enter a nursing home. In other words, the presene of asymmetri information is masked by heterogeneous risk attitudes or heterogeneous insurane demand. owever, their framework still fails to find private information even after introduing fators about individual heterogeneity, despite a diret evidene of private information when available. Fang, Keane and Silverman (2008) also provide evidene of advantageous seletion. hey find that having Medigap insurane would be assoiated with $4,000 less in total medial expenditure if not ontrolling for health status, and $2,000 more in total medial expenditure if ontrolling for health status. his result indiates that those who purhase Medigap insurane are healthier, providing evidene of advantageous seletion. 3

5 his paper makes three ontributions to the literature. First, we identify the reasons for failure to detet the private information in the urrent empirial framework even in the presene of the fators that are related to individuals heterogeneity of insurane demand. As disussed in F&MG, the presene of individual heterogeneity may ause the problem. owever, inluding fators that are related to individual heterogeneity may not solve the problem as long as we only have an inomplete set of fators that explain the individual heterogeneity. Seond, we present an alternative test for the private information in the presene of heterogeneous preferenes for insurane. We assume that individuals an be grouped into two types. he timid type has a stronger taste for insurane and is less likely to experiene the insured event. he bold type has a weaker preferene for insurane and a higher risk of experiening the insured event. he test of a positive orrelation between an individual s risk and the deision to purhase insurane is only valid if onditioning on either type. Sine types are not observed, we use the mixture density to jointly model the risk and insurane purhase. Several harateristis that are orrelated with the unobserved types are used to probabilistially determine whih type the person belongs to. he advantage of this method is that an inomplete set of variables that explain the individual heterogeneity is normally suffiient to produe onsistent estimates in the insurane demand and risk exposure equations, and to detet private information if it exists. he literature has been using the mixture density to identify unobserved types. Examples inlude Lee and Porter (1984), Keane and Wolpin (1997), Knittel and Stango (2003), and Gan and Mosquera (2008). enry, Kitamura, and Salanié (2010) explain onditions for identifiation of finite mixtures. hird, we apply this model to the sample of F&MG. he advantage of the F&MG sample is that the diret evidene of private information is available. We find that the two types of agents behave differently as predited. he timid are more likely to purhase insurane but less likely to enter the nursing home than the bold. Conditional on publi information and the type of an individual we obtain a statistially signifiantly positive orrelation between ex post risk and the 4

6 insurane purhases. his provides the evidene of the existene of private information. Allowing for two types of individuals makes it possible to detet the presene of private information. We onfirm the finding of F&MG without relying on diret evidene of private information. Moreover, we also onfirm that the timid type would be more likely than the bold type to purhase the insurane poliy but less likely to use insurane, as predited by the theory. Our method is most useful for insurane markets where suh diret evidene of private information is not omplete or even not available. It may unover the existene of private information while simultaneously aount for heterogeneity in risk attitudes and reveal the existene of asymmetri information without the diret evidene of private information. he paper is organized as follows, in setion 2 we illustrate the identifiation problem and explain how it is possible to aount for heterogeneity in risk preferenes. Setion 3 desribes the data and presents the results. Setion 4 onludes. 2. An Empirial Strategy to Detet Private Information in the Insurane Market We empirially haraterize the market for long-term are insurane (LCI) through two equations. he first equation relates individuals harateristis to the probability of entering a nursing home. he seond equation relates the same harateristis to the deision to purhase long-term are insurane. N 1 LCI 1 u 0 v 0 (1) We use to denote harateristis that are publi information information that is available to both the individual and the insurane provider. Individuals may have information about the likelihood of eventually entering a nursing home that is not available to the insurer. We denote this private information by. Without loss of generality we define so that β > 0. A higher probability of entering a nursing home reates an inentive to purhase long-term are insurane. As is not refleted in insurane premiums, this leads to a positive relationship 5

7 between and insurane purhases, δ > 0. Further, we inlude to denote individual taste for insurane or individual risk attitude. We define so that δ > 0, a higher value of implies a higher likelihood of purhasing insurane. At the same time, as pointed out by de Meza and Webb (2001), and supported by empirial evidene in F&MG and Fang, Keane and Silverman (2008), a higher value is assoiated with a lower level of ex post risk, i.e., β < 0. While both private information and individual heterogeneity are unobserved, they exhibit different effets on insurane purhase and ex post risk. he oeffiients for the private information, β and δ have the same sign in the equations haraterizing insurane purhase and ex post risk. he oeffiients for individual heterogeneity, β and δ, have opposite signs in the two equations. Finally, sine all ommon fators have already been onditioned in (1), the error terms u and v are assumed to be distributed standard normal u ~ N 0,1 and ~ 0,1 eah other, Cov(u, v) = 0. v N, and are independent on If it is not possible to observe or, the two equations an be estimated only partially: N 1 u 0 LCI 1 v 0 (2) he resulting error terms are given by: u u and v v. If individuals have homogeneous risk preferenes or insurane demand, there is no variation in. he orrelation between the error terms is given by: Cov u, v 0 1 (3) with β > 0 and δ > 0. herefore, the presene of unobserved private information leads to a positive orrelation, and estimating ρ 1 offers a way to empirially test for the presene of asymmetri information. owever, this test an fail to detet the presene of private information if individuals differ in their inlination to purhases insurane. With heterogeneous risk preferenes the orrelation of the error terms is desribed by: 6

8 7 ) ( ) ( ) ( 2, ambiguous v u Cov (4) he first term in (4),, is assumed to be positive in the presene of private information. owever, the seond term is negative if β and δ have opposite signs. In other words, if individuals with a low risk of nursing home use (β < 0) tend to have a taste for insurane (δ > 0), the orrelation of the two error terms in (4) is no longer indiative of the presene of asymmetri information, but rather a ombination of asymmetri information and heterogeneous taste in insurane. Further, the signs of the remaining two terms in (4) annot be determined without further assumptions. Without observing the sign of ρ 2 in (4) annot be determined ex ante. In general it is not possible to observe. owever, it may be possible to observe a set of variables W that are related to. F&MG propose suh variables: wealth, adoption of preventive health ativities, and seat belt usage. hey use these variables to estimate the following model, v W LCI u W N W W (5) If W fully haraterizes, the orrelation between the error terms an be used to test for private information. owever, if W only represents a subset of variables that haraterizes, the problem remains. For example, let be fully haraterized by observed W and unobserved M: =aw + M + ε, (6) then the error tem in the N model in (5) is u M u, and the error term in the LCI model in (5) is v M v. he orrelation between u and v is given by: ) ( ) ( ) ( 3 ambiguous M M M M (7) Again, similar to ρ 2 in (4), the sign of ρ 3 annot be determined ex ante without further assumptions. he first term in (7),, is assumed to be positive in the presene of private

9 8 information, and the seond term in (7), M M, is negative if β and δ have opposite signs. ut the signs of the remaining two terms in (7) annot be determined without further assumptions. herefore, it is possible that ρ 3 is not positive even if private information is present. owever, in the following disussion, we show that it is possible to solve this problem if we are willing to assume that the heterogeneity in the risk preferenes an be aptured by allowing eah individual to be one of two types with an individual speifi probability. In partiular, we assume that there are bold () or timid () individuals. takes two values, and. 4 Given this assumption, we an rewrite equations in (1) for eah type. For the timid type individuals ( = ), we obtain: v v v LCI u u u N (8) In both equations in (8), the effet of is absorbed into the onstant terms, and. Similarly, for the bold type individuals ( = ), we obtain: v LCI u N (9) Again, the onstant terms and absorb the effet of while the error terms inlude the private information. he model predits the relative magnitude of the onstant terms. Everything else equal, a timid type individual would be more likely to purhase LCI but less likely to enter the nursing home than a bold type individual, i.e. and. More importantly, the error terms u and v now only inlude the private information, but not the individual heterogeneity. herefore, the orrelation between u and v reflets the presene of private information. Imposing the two-type struture transforms the problem from one of 4 More generally, both and an be random variables that are unorrelated with and.

10 identifying β and δ in (1), to a problem of identifying and, and at least probabilistially the type of an individual. We need to jointly identify,,,,,, the probability of belonging to a ertain type, and the orrelations between the two error terms u and v. Conditional on and W, we observe four possible outomes, (N=i, LCI =j ), for i = 0, 1 and j = 0, 1. he probability of a given outome depends on the type of the individual: N i LCI j Pr N i, LCI j Pr Pr N i, LCI j Pr, Pr (10) As desribed in equation (6), is determined by W and M. We assume that the probability of being of a ertain type varies with W: W M Pr, Pr, for some W and M. (11) If we assume that W is observed but M is not observed, it is not possible to onsistently estimate the oeffiient in the equation below: Pr W, M PrW M 0 (12) Assuming that the unobserved M an be written as a linear funtion of W and an error term, i.e., M = Wα + τ ; equation (12) an be rewritten as Pr W PrW W 0 PrW 0 (13) where the random errors τ and ε are normally distributed, and the parameter γ is the sum of γ and α, saled by a onstant suh that ω ~ N (0, 1). In the linear model, γ may be onsistently estimated if M is unorrelated with W (where α = 0 ). owever, in the nonlinear setting here, γ annot be onsistently estimated regardless of the orrelation between M and W. he key identifying assumption is that onditional on the type of an individual W and M are not related to either the probability of entering a nursing home, or the probability to purhase long-term are insurane: 9

11 N i LCI j W M N i LCI j Pr,,, Pr, (14) Consequently, any assoiation between W and M and the probability of entering a nursing home or purhasing insurane is solely driven by the assoiation between W and M and the probability to belong to a ertain type. In a separate paper, enry, Kitamura and Salanié (2010) also propose this independene ondition as one of the key assumptions of identifying the model. Intuitively, this identifiation assumption is similar to the identifiation assumption of the instrumental variable model. W and M may be onsidered as the instrumental variables for the type variable. hey are assumed to be unorrelated with N and LCI but orrelated with. More generally, Pr N i, LCI j W, M PrN i, LCI j Pr W, M PrN i, LCI j Pr W, M (15) Again, the relative ontribution of W and M to the variation in does not affet Pr(N = i, LCI = j) onditional on the type of an individual. Rewriting (15) onditional on the observed variables and W gives: Pr N i, LCI j, W PrN i, LCI j, Pr W PrN i, LCI j, Pr W (16) All terms in (16) are defined in (8), (9), and (13). herefore, one may onstrut a likelihood funtion to estimate suh a model. Lemma: If the probability of being a ertain type varies with W (equation (12)), and onditional on the type of an individual W is not related to either the probability of entering a nursing home or the probability to purhase long-term are insurane (equation (14)), then estimating equations (8) and (9) together with either (12) or (13) produes onsistent estimates of the parameters,,,,,, and the orrelations between the two error terms u and v. 10

12 Equations (15) and (16) reveal that knowledge of M and estimation either (12) or (13) result in different estimates for the oeffiient of W, but the other oeffiients of the model are not affeted by the fat that M is not observed. he parameters of interest,,,,, β, δ and the orrelation between u and v remain to be onsistently estimated while oeffiients of W will not. If we have more than one dimension of information in W, we may only use a subset of W to estimate the model. his is very similar to the over identifiation test in the instrumental variable model where more than neessary instrumental variables are available. Similar disussions are also offered in enry, Kitamura, and Salanié (2010). In summary, the intuition of the method is very similar to the two-stage instrumental variable model. he onsisteny of the 2SLS estimates does not require the onsisteny of the first-stage regression. Similarly, the fat that we do not observe M does not reate inonsistent estimates of the parameters of interest. herefore, the advantage of the proposed method is that it only requires some (but not full) information about to identify the parameters of interest. Even the over identifiation test in the instrumental variable model has a orresponding test in the urrent model. In omparison, the method in the literature, as used by F&MG, replaes the unobserved type variable by a set of proxies W in the N and LCI equations. It works only if W ompletely haraterizes. owever, given how little we understand the unobserved heterogeneity, this is unlikely to hold in pratie. 3. Data and Results We illustrate our estimation proedure by applying it to the data assembled by F&MG. he data are based on the Asset and ealth Dynamis (AEAD) ohort of the ealth and Retirement Study (RS). his survey is designed to be representative of the non-institutionalized US population born in 1923 or earlier and their spouses. For more detailed information about sample and variables see F&MG. 11

13 It is possible to observe insurane status, nursing home utilization, and a number of demographi and health variables that make it possible to ontrol for risk lassifiation of individuals by insurers. F&MG apply an atuarial model used by many insurers to alulate a variable that reflets the ompany predition of nursing home use whih is used to determine premiums. his ompany predition aptures the available publi information,. he data also ontain information that is not used by insurers to set premiums. ased on a survey question, F&MG onstrut a measure of private beliefs about the likelihood of moving into a nursing home. We use the private believes as a proxy for private information,, apturing some but not all of the private information of individuals. he self-reported probability of entering nursing home has been shown to be onsistent on average with observed probabilities at the aggregate level, but has serious reporting errors at individual level (see, for example, urd and MGarry 2002; Gan, urd and MFadden 2005), suggesting that the measure an best serve a noisy proxy to the private information. he data also ontain information about wealth and proxies for risk attitudes. he proxies for risk attitudes are self-reported seat belt usage and whether individuals undertook preventative healthare measures, suh as flu shots or aner sreenings. able 1 displays the desriptive statistis. he sample ontains 5,119 individuals. 11% of them have long-term are insurane in 1995 and 16% enter a nursing home at some point from owever, to be omparable aross various speifiations, we limit our sample to individuals without missing information for any utilized variable. he working sample size onsists of 5,000 observations. We first estimate model (2) for only one type of agent to provide a baseline for our further analysis. In all speifiations, we ontrol in both equations for the publi information available to the insurane ompany,, summarized by the ompany preditor variable. 12

14 he estimates reported here are similar to those reported by F&MG. 5 In the first olumn in able 2, we onfirm that the ompany predition has a positive effet on the probability of entering a nursing home. he estimated oeffiient for the ompany preditor ( β ) is (0.090). 6 owever, the insurane ompany preditor redues the probability of purhasing insurane. he estimated oeffiient ( δ ) is (0.123), orresponding a marginal effet of We obtain a negative (not signifiantly different from zero) estimate at (0.041) for the orrelation between the two error terms. In other words the orrelation test does not provide evidene for the existene of asymmetri information. Next, in the seond olumn in able 2, we add the proxies for private information, individual preditions to enter nursing homes,, to the two equations. he oeffiient for this variable is positive in both equations, implying that indeed private information is present. he third olumn in able 2 displays the results for the model after we added proxies for the risk attitudes (types) of individuals, W, but without the individual predition. hese proxy variables for risk attitudes are dummy variables for preventative health measures taken, seat belt usage and for the 4 th, 3 rd, and 2 nd wealth quartile. We onfirm that the oeffiients for the variables in W have opposite signs in the two equations. F&MG argue that different signs in W in two equations atually indiate the heterogeneity in tastes. Now, we estimate the model with two types of individuals. We jointly estimate (13), (8), and (9). Let = 1 be the timid type and = 0 be the bold type. W onsists of seat belt usage, preventative healthare measures, and wealth quartiles. We restrit the oeffiients β, δ and the orrelation ρ to be idential for the two types, but allow the onstant terms differ. Column 1 in able 3 displays the results (the orresponding marginal effets are shown in able A2). he top panel illustrates the effet of fators prediting 5 o make our estimates omparable to eah other our speifiation differs slightly from those in F&MG. When using the exat speifiations as that of F&MG, we obtain the same results. 6 his orresponds to a marginal effet of 0.40 if other variables are evaluated at their means (See table A1). 13

15 the type of an individual. Overall, 28% of individuals belong to the timid type. Individuals of the timid type are haraterized by a higher inidene of preventative ativities and seat belt use; they also tend to be wealthier. A person who takes preventative ativities has a 11 perentage points higher probability to be the timid type. Always wearing seat belt inreases this probability by 13 perentage points. Finally, ompared with individuals in the fourth (lowest) wealth quartile, having a wealth level in the top quartile (first quartile) inreases the probability of belonging to the timid type by 28 perentages points, for individuals in the seond quartile the inrease is 20 perentage points, and for individuals in the third quartile it is 12 perentage points. People who belong to different types exhibit lear differenes in their behavior. As predited by the model, we find that a timid-type individual is more likely to purhase the LCI but less likely to enter a nursing home. For a timid-type person, the average likelihood of being insured is 0.41 while the hane to use nursing home is just One the ontrary, for the bold-type person, the average probability of purhasing long-term are insurane is less than 0.01; the odds of entering nursing home are he lower panel of able 3 displays the relationship between individual harateristis and insurane purhases and nursing home usage. As predited in the previous subsetion, the proposed model has lear preditions in terms of the relative magnitude of the onstant terms. For the LCI model, the estimated onstant for the timid type is (0.188), signifiantly larger than the estimated onstant for the bold type at (0.237). A one-sided -test rejets the null hypothesis that ˆ ˆ (p-value = 0.000). For the N model, the estimated onstant for the timid type is (0.215), statistially smaller than the estimated onstant for the bold type at (0.061), p-value = oth test results are onsistent with the preditions of the model. In terms of probabilities of purhasing LCI and entering nursing homes, the timid type would be 40 perentage points more likely to purhase LCI but 16 perentage points less likely to enter into nursing homes. 14

16 Most importantly, by separating individuals into two types, we are able to obtain lear evidene of private information. he estimated orrelation between the error terms in the N model and the LCI model is positive and statistially signifiant at (0.271). It is important to note that this is ahieved without using any data on private information. he seond set of estimates in able 3 inludes one dimension of private information, the individual predition of entering nursing homes, in both the LCI equation and the N equation. As in the ase of one type model in able 2, the oeffiient of this variable is positive in both LCI and N equations, showing the importane of suh private information in determining both the deisions to buy LCI and to enter nursing homes. Adding the proxy for private information redues the orrelation between the two error terms to (0.209), although it remains positive and statistially signifiant. his result reveals that (a) the individual predition of entering nursing home as eliited in the survey may only haraterize a small portion of the private information; and (b) adding more and more private information may eventually lead to zero orrelation, as predited by the model. Unlike the one-type model, our two-type model an detet the existene of the private information even in the absene of observable data on private information. If some data on private information is observed, our model an indiate to what extent additional unobserved private information influenes the deision to purhase insurane. As emphasized in the previous subsetion, similar to the over identifiation test in the instrumental variable model, one impliation of our model is that even a partial set of W may produe onsistent estimates of the key parameters of interest. herefore, as a further test of the model, we vary the hoie of variables in determining types, W. In our urrent setting, the set W onsists of wealth quartiles and preventive are, always wearing seat belts. Wealth is a natural andidate for W. As pointed out by F&MG Mediaid offers a better substitute for private insurane for low wealth individuals and is therefore orrelated with the risk preferenes of individuals. It is plausible that the variables preventative are and always wearing seat belt are 15

17 assoiated with risk preferenes, as well. owever, it might be argued that these variables are orrelated with the likelihood of eventual nursing home use, violating our identifying assumption. In the first two olumns in able 4, we do not inlude any variables in W. In olumn (1) we only inlude the ompany preditor,, while olumn (2) inludes individual private information on entering nursing homes,. he insurane ompany preditor remains positive in the N equation and negative in the LCI equation. More importantly, the onstant in both equations satisfy the preditions of our two-type model. owever, the orrelation between the error terms in both N equation and the LCI equation is no longer signifiant. his result is expeted sine there is no information to eonomially distinguish the two types of onsumers and the type is purely identified by funtional form. herefore, one of our two identifiation assumptions the probabilities of purhasing insurane and nursing home use have to vary with W is violated. he seond olumn inludes private information on entering nursing home. he oeffiient estimates for the private information variable are positive and signifiant in both N and LCI equations. Again, the orrelation between the error terms is not statistially signifiant. he third and fourth olumn in able 4 show the results when the wealth quartiles information is used. Our model shows that parameter estimates in both N and LCI equations should be similar to the orresponding parameter estimates in able 3 when all available information in W is used. oth sets of estimates are indeed similar to eah other. For example, in the olumn (3) in able 4, the oeffiient estimate for the insurane ompany preditor is (0.101) in the N equation. he orresponding oeffiient estimate in able 3 is (0.104). he oeffiient estimate for onstant term in N (timid type) equation in able 4 is (0.237) while the orresponding oeffiient estimate in able 3 is (0.215). he orrelation between the error terms in the N model and the LCI model is positive and statistially signifiant. It is (0.266) without any diret information on private information and as expeted drops to 0.472(0.275) after adding private information in olumn (4). 16

18 he fifth and sixth olumn in able 4 list results using variables of preventive ativities and seat belt usage with the fifth olumn only has the ompany preditor while the sixth olumn inludes both the ompany preditor and private preditor. he oeffiient estimates in both N equation and LCI equation have the expeted signs and are statistially signifiant. he estimates of the onstants are onsistent with the preditions of our model owever, the orrelation between the error terms in the N model and the LCI model is no longer positive. his highlights the importane of the validity of the identifying assumptions for the elements of W. As in a standard instrumental variable approah, a violation of the identifying assumptions will lead to biased results. able 5 presents a formal test. he test ompares the estimates of the parameters of interest in the N and LCI equations presented in able 3 and able 4. Our theoretial analysis suggests that having a partial set of W may affet the estimates for the oeffiient estimates for W in the type equation but not the oeffiients in the N and LCI equations. he first set of olumns in the table ompares estimates from the full model with the wealth-quartile-only model, while the seond set of olumns ompares estimates from the full model with the set of autious ativities (preventive ativity and seat belt use). he first row ompares the parameter estimates in both the N and LCI equations, while the seond row ompares the parameter estimates in the type-equations. As expeted, the estimates in the N and LCI equations in two models are not statistially different with eah other. Interestingly, the estimates in the type-determination equation from these two sets of estimates are not different from eah other, either. One potential reason for this may be due to fat that the omitted sets of W (the preventive ativity and seat belt use) are independent on the wealth quartiles. Finally, the seond panel in table 5 ompares the full model with the model with only prevention and seat belt usage variables as W. he χ 2 test statistis are very large, indiating that estimates from the two models are statistially different from eah other for both the oeffiients in the LCI and N equations and for the type equations. One possible reason is that 17

19 prevention and seat belt use variables may not be suffiient to identify the model. he true reason remains to be understood. 4. Conlusions Identifying private information in the insurane markets is important for empirially testing the eonomi theories of moral hazard and adverse seletion. It is also useful to improve effiieny of the insurane market. his paper proposes and estimates a new method to identify private information in the presene of heterogeneity onsumer types. We illustrate this method for an insurane market where diret evidene for private information is available. We are able to detet the presene of asymmetri information without using this diret evidene. In partiular, this paper makes three ontributions to the literature. First, when only a partial set of information is available to haraterize the individual heterogeneity, this study investigates the reason for the failure of urrent methods whih intend to test a positive relationship between the demand of insurane and the usage of the servie and identify the private information. Seond, based on the disussion, this paper proposes a new method to identify private information with individuals risk heterogeneity only using a partial set of information to haraterize suh risk heterogeneity. his method assumes that individuals risk heterogeneity an be grouped into two unobserved ategories. hird, although private information is known to be present in the long term are market, it annot be deteted by the existing method. owever, it an be deteted by the proposed method. he identifiation of this method is similar to that of the instrumental variable model. It requires the variables haraterizing risk heterogeneity are onditional on the type of an individual unorrelated with deisions to purhase insurane and to use the servie overed by the insurane. he proedure desribed here is a tool that an be used to detet the presene of private information in the insurane markets with heterogeneous risk preferenes even when the diret evidene of private information is not available. 18

20 19

21 Referenes Cawley, John and Philipson, omas (1999). An empirial examination of information barriers to trade in insurane. Amerian Eonomi Review, 1999, 89 (4), pp Chiappori, Pierre-Andre (2000). Eonometri models of insurane under asymmetri Information. In Georges Dionne, ed. andbook of Insurane Eonomis. London: Kluwer. Chiappori, Pierre-Andre and Salanié, ernard (2000). esting for asymmetri information in insurane markets. Journal of Politial Eonomy, 2000, 108 (1), pp Chiappori, Pierre-Andre, Jullien, runo, Salanié, ernard and Salanié, Franois (2006). Asymmetri information in insurane: general testable impliations. Rand Journal of Eonomis. Vol. 37, No. 4 (Winter, 2006), pp Cohen, Alma (2005). Asymmetri information and learning: evidene from the automobile insurane. he Review of Eonomis and Statistis. May (2): de Meza, David and Webb, David C. (2001) Advantageous seletion in insurane markets. Rand Journal of Eonomis, Summer (2), pp Finkelstein, Amy and MGarry, Kathleen (2006). Multiple dimensions of private information: evidene from the long-term are insurane market. Amerian Eonomi Review, Vol. 96 No.4, Page Finkelstein, Amy and Poterba, James (2002). Seletion effets in the market for individual annuities: new evidene from the United Kingdom. Eonomi Journal, 2002, 112 (476), pp Fang, anming, Keane, Mihael P. and Silverman, Dan (2008). Soures of advantageous seletion: evidene from the Medigap insurane market. Journal of Politial Eonomy. Volume 116, Issue 2, Page , Apr 2008 e, Daifeng (2009). he life insurane market: asymmetri information revisited. Journal of Publi Eonomis. enry, Mar, Kitamura, Yuihi, and Salanié, ernard (2010) Identifying Finite Mixtures in Eonometri Models Cowles Foundation Disussion Paper No urd, Mihael D. and MGarry, Kathleen (2002). "he Preditive validity of subjetive probabilities of survival." Eonomi Journal, 2002, 112(482), pp Gan, Li, urd, Mihael D. and MFadden, Daniel (2005). "Individual subjetive survival urves," in David Wise, ed., Analysis in eonomis of aging. Chiago: University of Chiago Press, 2005, pp

22 Gan, Li and Mosquera, Roberto (2008). An empirial study of the redit market with unobserved onsumer types. NER WPS # Keane, Mihael P. and Wolpin, Kenneth I. (1997) he areer deisions of young men. Journal of Politial Eonomy, Vol. 105, No. 3 (Jun., 1997), pp Knittel, Christopher and Stango, Vitor (2003). Prie Ceilings as Foal Points for ait Collusion: Evidene from Credit Cards. Amerian Eonomi Review, Vol. 93, No. 5 (De., 2003), pp Lee, Lung-Fei and Porter, Robert (1984). Swithing regression models with imperfet sample separation information: appliation on artel stability. Eonometria, Vol. 52, No. 2 (Mar., 1984), pp Rothshild, Mihael and Stiglitz, Joseph (1976) Equilibrium in ompetitive insurane markets: an essay on the eonomis of imperfet information. Quarterly Journal of Eonomis, 1976, 90 (4) pp Smart, Mihael (2000) Competitive insurane markets with two unobservables. International Eonomi Review, 2000, 41(1) pp

23 able 1: Desriptive Statistis Variable Mean sd Min Max Nursing ome Use Long erm Care Insurane Insurane ompany predition Individual predition Preventive health ativity Always wear seat belt op quartile of wealth rd Wealth quartile nd Wealth quartile Note: he sample onsists of the elderly aged 78 on average in 1995 who reported long-term are insurane status and nursing home use from 1995 to 2000 from the Asset and ealth Dynamis (AEAD) ohort of the ealth and Retirement Study (5,119 observations). 22

24 able 2: One type ivariate Probit Model Company predition Company and individual predition Company predition, prevention and wealth quartiles Company and individual predition, prevention and wealth quartiles (1) (2) (3) (4) N Insurane ompany predition (0.090) (0.091) (0.093) (0.094) Individual predition (0.092) (0.092) Preventive health ativity (0.081) (0.081) Always wear seat belt (0.056) (0.056) op quartile of assets (0.072) (0.072) 3 rd Wealth quartile (0.071) (0.071) 2 nd Wealth quartile (0.071) (0.071) Constant (0.034) (0.037) (0.081) (0.082) LCI Insurane ompany predition (0.123) (0.126) (0.128) (0.131) Individual predition (0.094) (0.097) Preventive health ativity (0.095) (0.096) Always wear seat belt (0.068) (0.068) op quartile of assets (0.088) (0.089) 3 rd Wealth quartile (0.090) (0.091) 2 nd Wealth quartile (0.093) (0.094) Constant (0.034) (0.038) (0.112) (0.116) Correlation of two error terms ρ (0.041) (0.040) (0.041) (0.041) Number of observations 5,000 5,000 5,000 5,000 Log-Likelihood Notes: Estimation of a bivariate probit of any nursing home use ( ) and long-term are insurane overage (1995).,, denote statistial signifiane at the 1%, 5%, and 10% level, respetively. Our estimates are weighted using the 1995 household weights. 23

25 able 3: wo-type Model Company preditor only Company and Individual preditors (1) (2) ype (timid type = 1) Preventive health ativity (0.145) (0.151) Always wear seat belt (0.104) (0.104) op quartile of assets (0.132) (0.133) 3 rd Wealth quartile (0.133) (0.133) 2 nd Wealth quartile (0.133) (0.134) Constant (0.185) (0.183) imid type old type imid type old type N Insurane ompany predition (0.104) (0.104) (0.103) (0.103) Individual predition (0.098) (0.098) Constant (0.215) (0.061) (0.239) (0.063) LCI Insurane ompany predition (0.184) (0.184) (0.190) (0.190) Individual predition (0.197) (0.197) Constant (0.188) (0.237) (0.166) (0.331) ρ (0.271) (0.209) Loglikelihood Number of Obs 5,000 5,000 Notes: Estimation of a two-type model of any nursing home use ( ) and long-term are insurane overage (1995).,, denote statistial signifiane at the 1%, 5%, and 10% level, respetively. Our estimates are weighted using the 1995 household weights. 24

26 able 4: wo-type Model: Robustness hek Constant Wealth quartiles Preventive ativity and Seat belt (1) (2) (3) (4) (5) (6) Pr(timid type = 1) Preventive health ativity (1.676) (1.636) Always wear seat belt (1.331) (1.390) op quartile of wealth (0.136) (0.140) 3 rd Wealth quartile (0.129) (0.132) 2 nd Wealth quartile (0.124) (0.126) Constant (0.052) (0.051) (0.189) (0.178) (1.871) (1.898) N (timid type) Insurane ompany predition (1.462) (1.409) (0.101) (0.105) (0.091) (0.092) Individual predition (0.209) (0.097) (0.092) Constant (1.301) (1.255) (0.237) (0.396) (0.057) (0.061) N (bold type) Insurane ompany predition (1.462) (1.409) (0.101) (0.105) (0.091) (0.092) Individual predition (0.209) (0.097) (0.092) Constant (0.123) (0.129) (0.061) (0.066) (0.053) (0.055) LCI (timid type) Insurane ompany predition (0.124) (0.127) (0.213) (0.215) (0.125) (0.128) Individual predition (0.094) (0.275) (0.096) Constant (0.057) (0.058) (0.273) (0.209) (0.059) (0.059) LCI (bold type) Insurane ompany predition (0.124) (0.127) (0.213) (0.215) (0.125) (0.128) Individual predition (0.094) (0.275) (0.096) Constant (0.098) (0.100) (0.256) (0.591) (0.080) (0.082) ρ (0.114) (0.113) (0.266) (0.275) (0.042) (0.042) Loglikelihood Number of Obs 5,000 5,000 5,000 5,000 5,000 5,000 Notes: he same as able 3. 25

27 able 5: ausman test: aseline model vs Robust hek aseline model vs wealth only aseline model vs Preventive ativity & Seat belt only (1) (2) Company predition oth preditions Company predition oth preditions N and LCI equations (0.998) (0.999) (0.000) (0.000) ype equation (0.768) (0.706) (0.027) (0.029) Notes: able reports the ausman test statistis and p-values in the parenthesis. 26

28 N able A1: One type ivariate Probit Model: Marginal effets at average Company predition (1) Company and Individual predition (2) Company predition and wealth (3) Company and Individual predition and wealth (4) Insurane ompany predition (0.021) (0.021) (0.021) (0.021) Individual predition (0.020) (0.020) Preventive health ativity (0.018) (0.018) Always wear seat belt (0.012) (0.012) op quartile of assets (0.016) (0.016) 3 rd Wealth quartile (0.016) (0.016) 2 nd Wealth quartile (0.016) (0.016) LCI Insurane ompany predition (0.023) (0.023) (0.023) (0.023) Individual predition (0.017) (0.017) Preventive health ativity (0.017) (0.017) Always wear seat belt (0.012) (0.012) op quartile of assets (0.015) (0.015) 3 rd Wealth quartile (0.016) (0.016) 2 nd Wealth quartile (0.016) (0.016) Number of observations 5,000 5,000 5,000 5,000 Notes: able reports marginal effets of marginal suess probability of entering nursing home and long-tern are insurane overage from bivariate probit estimation of equation (2) and (5).,, denote statistial signifiane at the 1%, 5%, and 10% level, respetively. Our estimates are weighted using the 1995 household weights. 27

29 able A2: wo-type Model: Marginal Effets Company preditor only Company & Individual preditors ype (timid type = 1) Preventive health ativity (0.062) (0.061) Always wear seat belt (0.056) (0.057) op quartile of assets (0.099) (0.100) 3 rd Wealth quartile (0.080) (0.080) 2 nd Wealth quartile (0.062) (0.062) imid type old type imid type old type N Insurane ompany predition (0.307) (0.206) (0.306) (0.207) Individual predition (0.068) (0.042) LCI Insurane ompany predition (0.040) (0.216) (0.053) (0.239) Individual predition (0.196) (0.059) Number of Obs 5,000 5,000 Notes: able reports marginal effets of marginal suess probability of entering nursing home and long-tern are insurane overage from two-type model of equation (8), (9) and (13). For the type equation, we estimate the marginal suess probability of being timid type.,, denote statistial signifiane at the 1%, 5%, and 10% level, respetively. Standard errors are alulated by Monte Carlo simulation. 28