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Disentangling adverse selection, moral hazard and supply induced demand: An empirical analysis of the demand for healthcare services. Vincenzo Atella ...

CEIS Tor Vergata RESEARCH PAPER SERIES Vol. 14, Issue 10, No. 389 – June 2016

Disentangling adverse selection, moral hazard and supply induced demand: An empirical analysis of the demand for healthcare services Vincenzo Atella, Alberto Holly and Alessandro Mistretta

This paper can be downloaded without charge from the Social Science Research Network Electronic Paper Collection http://papers.ssrn.com/paper.taf?abstract_id=2801679

Electronic copy available at: http://ssrn.com/abstract=2801679

Electronic copy available at: http://ssrn.com/abstract=2801679

Disentangling adverse selection, moral hazard and supply induced demand: An empirical analysis of the demand for healthcare services by Vincenzo Atellaa , Alberto Hollyb and Alessandro Mistrettac

May 2016

Preliminary version. Please do not quote without the authors’ explicit consent.

a

CEIS Tor Vergata, Department of Economics and Finance, Universit` a di Roma Tor Vergata Rome, Italy; CHP-PCOR, Stanford University

b

Institute of Health Economics and Management (IEMS), Faculty of Business and Economics University of

Lausanne, Switzerland; Nova School of Business and Economics, Universidade Nova de Lisboa, Lisbon, Portugal; Department of Engineering and Applied Sciences, University of Bergamo, Italy. c

Bank of Italy and Universit` a di Roma Tor Vergata.

Keywords: Quadrivariate probit, FIML, Supply induced demand, Moral hazard, Adverse selection, Health insurance. JEL classification: I13, I11, D82, C35

Electronic copy available at: http://ssrn.com/abstract=2801679

Abstract In the healthcare sector, Adverse Selection (AS), Moral Hazard (MH) and Supply Induced Demand (SID) are three very important phenomena that affect patients’ behaviour. Despite there exists a vast theoretical and empirical literature on these phenomena, so far, no contribution has been able to approach them jointly. This is mostly due to the difficulty to model the joint determinants of health service utilisation and health insurance choice by means of a tractable structural simultaneous equation model. In this paper, we provide a solution to this problem and estimate a simultaneous four equation structural model with four latent variables, where the first two equations are meant to deal with the adverse selection issue, while the third and fourth equation deal with moral hazard and SID issues. A closed form solution for the likelihood function - which guarantees an exact solution - is maximised by the means of FIML, using a large cross-sectional dataset from the Italian healthcare system. Empirical analysis has confirmed the theoretical predictions of our structural model. In particular, we find evidence of AS in the choice of private insurance and SID, but do not find MH behaviour on the patient side. These results are extremely important from a health policy perspective, given the existing debate on the development of a second pillar in the financing of the healthcare system in Italy and Europe.

Electronic copy available at: http://ssrn.com/abstract=2801679

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1. Introduction Since the seminal papers by Pauly (1968) and Rothschild and Stiglitz (1976), economists have largely investigated how asymmetric information in the insurance markets can produce inefficient outcomes, notably overconsumption, due to the insurgence of moral hazard (MH) and adverse selection (AS) behaviour by the individuals who buy more insurance if they expect to use more medical care (different risk types). Within this sector, the health insurance market represents a particularly interesting setting, where, to explore these issues, given the role that information asymmetry play between patients and physicians, and the well-known phenomenon of rising healthcare expenditure that characterises this sector. This last aspect is particularly relevant for economic and financial sustainability of the government and employer-provided insurance plans, giving rise to a considerable academic and public policy debate, aimed to better understand the role of selection and moral hazard, particularly in this context. From a policymaker’s point of view, disentangling these two effects is extremely important, in order to to design effective policies since the implications, in terms of regulation policies are quite different. According to Cutler and Zeckhauser (2000), and, more recently, to Powell and Goldman (2016), it is the relative importance of adverse selection vs. moral hazard that should drive the design of optimal policies, given that the policy instruments to address selection are different from those required to address moral hazard: in the first case, “risk pooling” may be the best choice, while in the second case “cost-sharing” is more suitable. In fact, while the first effect deals with the presence of additional insurance, the second effect is related to consumption decisions and implies that insured individuals consume more than the uninsured. Moreover, from an empirical point of view, if selection is an important phenomenon, then, the estimates of moral hazard obtained by neglecting selection are upward biased, and could lead to overestimation of the effect of policy instruments that are used to address moral hazard (Gardiol et al., 2005a). Complexity in this setup can be further increased if we consider that MH is, in turn, the sum of two components: “demand-side” moral hazard and “supply-side” moral hazard or Supply Induced Demand (SID) behaviour. The latter could be defined through Bickerdyke et al. (2002) as “the notion that doctors, in acting as agents for their patients, can use their discretionary

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power to engage in demand-shifting or inducement activities such that their recommended care differs from that which an informed agent would deem appropriate”.1 Given that AS and both types of MH are positively correlated with the relationship between ex-post realisation of risk and insurance coverage (Chiappori and Salanie, 2000), disentangling these three effects is not an easy task if the researchers observe only the relationship between the presence of a supplementary health insurance and the use of medical care (Chiappori and Salanie, 2000; Chiappori, 2000; Finkelstein and Poterba, 2004). From an empirical perspective, literature has usually approached the problem leaving self-selection and moral hazard as distinct phenomena. As recognised by Einav et al. (2013), the early seminal theoretical contributions in this field have abstracted from moral hazard and have rather focused on selection, driven by onedimensional heterogeneity in risk aversion (Finkelstein and McGarry, 2006; Cohen and Einav, 2007; Fang, Keane, and Silverman, 2008; Einav, Finkelstein, and Schrimpf, 2010). On the contrary, as long as data from large randomised experiments were available, control for endogenous insurance selection was possible, and moral hazard effects were correctly estimated. Various studies have used such design. A well-known early example is RAND Health Insurance Experiment (RAND HIE) (Manning et al. 1987; Newhouse and the Insurance Experiment Group, 1993; Newhouse et al. 2008). It was started in 1971 and was a multi-year, 295 million (in 2011 dollars, (Greenberg and Shroder, 2004)) medical care study, that, among other things, randomly distributed health insurance plans to the participants in 6 U.S. cities and recorded their health and medical care consumption in the following years. Another example is the Oregon Health Insurance Experiment (Oregon HIE) (Finkelstein et al., 2012; Baicker et al., 2013). It was started in 2008 by the Oregon Health Authority, who expanded the state’s Medicaid program to 10,000 additional low-income adults using a lottery. In both cases, the random assignment allowed the 1

According to Zweifel et al. (2000), SID occurs because i) Providers have an informational advantage over patients, which could be used to alter patient demand functions, whenever it is unlikely that the medical treatment harms the patient; ii) Patients hold supplementary health insurances which limit patient out-of-pocket expenditure: the lower is patient co-payment, the higher is provider ability to induce demand. In these cases patients could be reluctant to fill the asymmetry information gap (which is a costly activity) and more prone to accept the extra health care services proposed by physicians; iii) Physician have incentive to create his own demand. This idea is formalized by Evans (1974), who assumes that the physician’s utility depends positively on income and negatively on the level of inducement. In other words, the physician is only willing to induce demand if this leads to a higher income. Clearly, whenever these situations occur jointly, the probability to observe SID effect is higher.

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researchers to study moral hazard without concerns for endogenous insurance selection. Unfortunately, with this approach, researchers can lose important pieces of information. In fact, a potential drawback inherited in randomisation is where it removes the endogeneity aspect of the choice and, as noticed, “it thus abstracts, by design, from any selection on moral hazard, which could have important implications for the spending reductions achieved through offering plans with higher consumer cost sharing.” (Einav et al., 2013, p.179) Health economics literature emphasized that the demand for health care is conditioned by the health insurance status of the user and that the insurance decision itself will depend upon, inter alia, expected future consumption of health services, implicitly assuming the coexistence of moral hazard and adverse selection. Several researchers have attempted to model this interdependence by developing microeconomic structural models of joint demand for health insurance and healthcare. The framework for most of these models is somewhat similar. The interdependence of the demand for health insurance and healthcare, given that future health status are uncertain, is generally modelled as a two-stage decision which is made by an expected-utilitymaximizing consumer. In the first stage, the consumer chooses the insurance policy that yields the highest expected utility. In the second stage, after the uncertain health state is realized, the consumer chooses healthcare consumption. Examples are Cameron et al. (1988), Cardon and Hendel (2001), Gardiol, Geoffard and Grandchamp (2005a, 2005b) and Einav et al. (2013), but this list is far from being exhaustive. At the econometric level, the critical task is to distinguish between the relative importance of the moral hazard problem and self-selection while determining the demand for health services. Except Cardon and Hendel (2001), the papers mentioned above conclude with the co-existence of moral hazard and adverse selection, resulting in the endogeneity of insurance choice in demand for healthcare equation. In particular, Gardiol, Geoffard and Grandchamp (2005a, 2005b) showed that in the context of the Swiss health insurance system, where all individuals are offered the same menu of contracts (which includes a choice of deductibles), the negative association between mortality risk and coverage (deductibles) indicates that individuals do not select their deductibles at random. Their results indicate that the observed 33.6% difference in outpatient expenditures between those with the largest and smaller deductible may be roughly attributed to partly (17.1%)

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on the selection effect, and the other half (16.5%) on the incentive effect. A similar conclusion is reached by Einav et al. (2013) which show, at least in their context, abstracting from selection effect could lead to overestimation of spending reduction which is associated with introducing a high-deductible health insurance option. The decision procedure modelled in the papers mentioned above leads to a two-stage sequential microeconomic structural model. An alternative empirical strategy consists of directly settingup a simultaneous-equation econometric model to jointly address the possible self-selection and demand-side moral hazard. A prototype is a simple simultaneous two-equation model which contains two endogenous variables. The first is a latent variable representing the propensity of an individual to buy a supplemental insurance, while we only observe the dichotomous variable associated with it. The second endogenous variable, that we assume to be observable, represents healthcare expenditures or utilization. The first equation is a reduced form equation of the insurance choice made by the individual, while the second equation is a structural equation related with the demand for healthcare services, conditional on the type of insurance plan he/she has selected. The dummy variable contained in the second equation may be endogenous if the two endogenous variables are correlated, which corresponds to the self-selection (adverse selection) effect. Thus, testing for selection is tantamount to test for endogeneity of the dummy variable associated with insurance choice in the equation explaining health expenditure. On the other hand, moral-hazard corresponds to the regression coefficient of the dummy variable (endogenous or not) being different from zero. Thus, this simple joint decision two-equation model, in the spirit of the selection models discussed by Maddala (1985), allows with disentangling demand-side moral hazard and self-selection effects. This basic prototype can be further modified and extended to various types of specific models. An example of a simultaneous two-equation model is provided by Holly et al. (1998), where, as before, the first equation is a reduced form equation of insurance choice, while the second is a structural equation for the propensity that an individual has at least one inpatient stay, given that he/she has used some medical treatment and conditional on the type of insurance plan he/she has selected. Given that the price-elasticity of “inpatient” services is usually found to be close to zero (Newhouse and the Insurance Experiment Group, 1993), moral hazard in the demand

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for inpatient services should only be interpreted as a supply-side moral hazard (SID). Note that, contrary to the two-equation model discussed earlier, the second equation is defined only as a subsample of the population. Therefore, the two-equation model, discussed in Holly et al. (1998) may be defined as a sequential model as in Maddala (1985), or as an endogenous sample selection model. One could further extend the basic prototype model trying to disentangle “demand-side” moral hazard (MH) and “supply-side” moral hazard (SID). One way to achieve this goal is to disaggregate the demand for healthcare services according to different price-elasticities, like, for example, “outpatient” and “inpatient” services. In this case, one can consider a simultaneous three-equation model where the first equation represents insurance choice like before and introduce two demand equations for health services: one for “outpatient” services, and another one for “inpatient” services, in the spirit of Gardiol, Geoffard and Grandchamp (2005a, 2005b). Although in this case we can potentially identify SID for “inpatient” services, nothing can be said about the relative importance of MH and SID. To partly solve this problem, for the first time, we estimate a simultaneous four-equation model with four latent variables and two selection processes, where the first two equations are meant to deal with the AS issues, while the last two equations deal with the disentangling of MH and SID (similar to the second equation of Holly et al., 1998). In particular, the first equation is a structural one provided for the propensity that an individual has any type of access to healthcare services, while the second equation is again a reduced form equation of insurance choice, conditional on having any type of access to the healthcare services. Then, the third equation investigates the demand for private hospital services (the more sensitive to private insurance effect), which is conditional on holding a private insurance, while the fourth analyses the sole use of “invasive” private hospital services, conditional on having had access to private hospital services. Within this new setup, the identification of MH and SID is possible for the following reasons: i) restricting the demand to “invasive” procedures where we are extremely confident that (demand side) MH aspects are ruled out ; ii) as “invasive” procedures are a subset of the private hospital services, comparing the MH parameters of the third and fourth equation can give us a good understanding of the relative role that MH and SID can have in private hospital services; In conclusion, although

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this model does not allow to obtain full understanding of the relative importance of MH and SID in all sectors of the Italian healthcare services, it represents, by far, a big improvement with respect to the previous literature. The simultaneous four-equation model proposed here is again a model with a mixed structure in the terminology suggested by Maddala (1983). It contains endogenous latent variables as well as an endogenous dummy variable. We estimate it through the Full Information Maximum Likelihood procedure (FIML) which numerical implementation presents important difficulties. In fact, computation of the probabilities which would appear in the likelihood function requires integration of the multivariate normal density over infinite range subsets of two, three and four dimensional real spaces. One of the important novelties of the paper is the numerical computation of these probabilities by means of a new decomposition of the above mentioned integrals which are developed by Huguenin (2004), extending the original work of Lazard-Holly and Holly (2002).2 From a numerical point of view, the main advantage of this decomposition is to transform multiple infinite range integrals of the multivariate normal density into a sum of finite range integrals, where the domains of integration are bounded and delimited by the correlation coefficients. Numerical computation of these probabilities and the derivatives of the log-likelihood function with respect to the parameters of the model thus are based on the exact analytical formulas. An alternative method would be to use routines that estimate the multivariate probit using simulation methods as, for example, the routine proposed by Roodman (2011), which provides the possibility to estimate a class of models like the one proposed in this paper using Simulated Maximum Likelihood (SML) techniques (cmp Stata command). However, the exact numerical computation procedure is preferred, since, according to Huguenin (2004) and Huguenin et al. (2014), the estimates obtained with this FIML procedure seem to be more accurate and more efficient than those based on SML methods. A comparison between these two types of numerical procedures in the context of our model is discussed in this paper. The empirical analysis is conducted using a large Italian cross-sectional survey from the 1999 Multiscopo Survey (MS) (ISTAT, 2001). Our results show that: i) in the Italian private health insurance market, we observe adverse selection problems like the propensity of a person to receive 2

See also Huguenin et al. (2014) for an exposition of the main results of Huguenin (2004).

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any medical service may not be considered as exogenous in the equation, explaining the propensity of the same person to select a supplementary health insurance plan; ii) when looking at any type of healthcare service demanded, we observe moral hazard as well, but, in this case, we are not able to disentangle SID from “patient induced” moral hazard; iii) insured individuals with supplementary health insurance have a higher probability to access any private health service (moral hazard), given that he/she has used some medical treatment; iv) however, this last effect is solely driven by provider SID, given that we do not find evidence of “patient induced” moral hazard. We believe that these results are particularly reliable, given that our healthcare framework is greatly simplified when compared to other countries. In fact, in Italy, the presence of a well-functioning and effective universalistic public health care system greatly simplifies the model’s theoretical assumptions, as we do not have to care about individuals without insurance or those who tradeoff between public and private insurance. The paper is organised in seven sections, including this Introduction. Section 2 presents some background information concerning the Italian health care system. Section 3 introduces the econometric model and discusses coherency, as well as the conditions for identification. Section 4 describes the data used and presents some descriptive statistics concerning the use of private health insurance and the related use of private healthcare services. Section 5 presents the empirical results, while in Section 6 we add some robustness checks. Finally, Section 7 discusses the policy implications of these findings and draws some conclusions.

2. The healthcare system in Italy and the private health insurance demand. Since 1978, the Italian healthcare system has been regulated and administered through the National Health Service (Servizo Sanitario Nazionale, referred to henceforth as SSN). The system is based on four main principles: universal coverage, full range of the health services provided, participation of citizens in the management of the SSN, and organizational pluralism (State, Regions and Local Health Authorities). The main objective of the system is then to guarantee everyone equal access to uniform levels of healthcare, according to their needs. Income, personal or social characteristics or geographical location ought not to influence the level of coverage.

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The system is organized in to three hierarchic levels: national, regional and local units. The national level is responsible for designing the National Health Plans with the aim to ensure the general health objectives and interventions. It is then the responsibility of the regional governments to achieve the objectives posed by the National Health Plan. Regions deliver the benefit package to the population through a network of population-based healthcare organizations (Local Health Units) and public and accredited private hospitals. Each region plans healthcare activities and organizes the supply according to population needs. Moreover, they have the responsibility to guarantee the quality, appropriateness and efficiency of the services provided. Local Health Units ought to guarantee equal access, efficacy of preventive, curative, rehabilitation interventions and efficiency in the distribution of services. They are also responsible for the balance between the funding provided by regions and the expenditures for the provision of healthcare services. In providing healthcare services, the SSN uses a wide array of providers: hospitals with different forms of ownership (directly managed, public hospital trusts, public and private teaching hospitals, public and private research hospitals, non-profit hospitals, for profit hospitals) GPs, public and private ambulatory care facilities (specialist and diagnostic), public and private rehabilitation facilities and private community nurses. Physician services are provided by both General Practitioners (GPs) and specialists, through a referral system. GPs are paid on a capitation basis. Patients do not pay for visits to the GPs, and there is no limit to the number of visits they can have. Concerning specialists, there are public (SSN) and private ones. Private specialists are, generally, self-employed and, usually, they work in “single practices” ambulatory while, very rarely, we find two or more specialists with different specializations in the same office. For each visit to a specialist in the public system, patients pay a fee (about 40 euros, quite low compared to the average fees in the private sector), but access is regulated by the GPs. Visits to private specialists are unregulated. Similarly, there is a maximum fee of 40 euros for diagnostic tests done in public facilities, while no charge is requested for hospitalization3. Co-payments depend on the income, age and health conditions of the patients and are required for drugs, ambulatory treatments, specialists and also for some diagnostic and laboratory tests. Public healthcare system coexisted with private provision of medical services. Private financing 3

Co-payment fee and access to specialists may vary across regions due to different co-payment policies adopted

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plays an increasing and not so marginal role. In this case, about 90 percent of private expenditure is financed by the patient’s out-of-pocket payments, with the remaining 10 percent covered by private health insurance and company health plans. Furthermore, according to the 1999 Multiscopo survey by the Italian National Institute for Statistics (ISTAT), the share of Italian population covered by private health insurance is about 10 percent, well below what is recorded in other EU countries. This low share comes, despite the fact that after the health reforms occurred between 1992 and 1994, the household private health expenditure has reached a level which is close to 30 percent of the overall healthcare expenditure, thus posing Italy as one of the industrialized countries with the highest use of the so called out-of-pocket health expenditure. Presently, the insurance market proposes both individual and collective insurance policies. Individual insurance policies are mainly structured as payment of a daily allowance for each sick leave and they are typically subscribed by self-employed workers with a high income, with the aim to reduce economic damage from illness; collective policies grant a substitute coverage (as a matter of fact, it is a duplication of the public healthcare system): usually they are restricted to executives or high income workers - who pay twice - in order to obtain healthcare services with more comfort. Private utilisation of healthcare services is strongly influenced by two main factors: waiting lists and perceived low quality of healthcare services. Recently, some public hospitals have been organised in a way to offer “hotel” services (mainly single rooms with better services) to those patients willing to pay for out-of-the-pocket service. In this last case, although being hospitalised in a public facility, private insurance can help afford such costs.

3. The econometric model As partly discussed in the introduction, healthcare utilisation and private health insurance are strongly interrelated, giving rise to three different effects: adverse selection (AS), moral hazard (MH) and supply induced demand (SID). In particular, those who expect to use more healthcare services are more likely to buy supplementary insurance, thus, causing problems of AS (or self selection). At the same time, MH intervenes each time patients see their out-ofthe-pocket expenditure reduced, due to the presence of supplementary private insurance. Finally,

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patients with supplementary private insurance may experience a higher exposition to “unwanted” medical care - or SID. Therefore, in the presence of health insurance patients, the choices with respect to these three phenomena are not independent, nor it is easy to estimate and disentangle their effects on the demand for healthcare services, based on the information on supplementary insurance subscription and the utilisation of healthcare services. In order to better understand and to measure these relationships, in this section, we introduce a theoretical framework, making use of a simple four equation structural model. In a more formal way, let y1∗ be an endogenous variable representing the propensity of a person to receive any medical service, y2∗ an endogenous variable representing the propensity of the same person to purchase a supplementary health insurance, while y3∗ and y4∗ are the propensity the same person to receive any type of healthcare service from private hospitals and any type of “invasive” healthcare service from private hospitals, respectively.4 To sketch our model, we start assuming that the individuals make predictions about their future health status and, therefore, on their propensity to receive any medical service (y1∗ ), which is conditional on a set of exogenous variables (x1 ). We then assume that individuals decide whether to buy (or not) supplementary insurance (y2∗ ) conditional on y1∗ and on a set of exogenous variables (x2 ). This is equivalent to state that y2∗ is simultaneously determined by y1∗ and a set of exogenous variables x2 . As such, these two equations are meant to deal with the AS issue. To deal with MH and SID effects, we need to further assume that, after making that choice, once the person is ill, his/her propensity to use privately provided healthcare services (y3∗ , y4∗ ) depend on the presence of additional insurance coverage (y2∗ ) and on personal characteristics (x3 , x4 ). Unfortunately, this assumption alone does not allow to disentangle between MH and SID, as y2 is responsible for both effects (in a positive way) on the demand for private healthcare services. One possibility to disentangle these two effects is to identify a subset of healthcare services affected either by MH or SID. In our case, we decided to look at the subset of services that we defined as “invasive” (y4∗ ), which, by their nature, should not be subject to “demand induced” MH effect. Therefore, we define a fourth equation where y4∗ is determined by y2 and by a set of exogenous variables x4 . Therefore, the 4We define as “invasive” procedures all sort of treatments that may cause a substantial discomfort to patients in

terms of both pain and potential side effects. In particular we consider as invasive treatments surgery interventions, gastroscopies and colonoscopies.

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occurrence of y1∗ is a precondition for y3∗ and the occurrence of y3∗ is precondition for y4∗ . As a result, this model has two selection processes and an endogenous variable and it falls in the category of what Amemiya (1975) called “sequential models”. The structural form of the model may thus be written as follows

y1∗ = x′1 β10 + u01

(3.1)

0 ∗ y2∗ = x′2 β20 + α21 y1 + u02

(3.2)

0 y2 + u03 y3∗ = x′3 β30 + γ32

(3.3)

0 y4∗ = x′4 β40 + γ42 y2 + u04

(3.4)

where 

u01



     u0   2  D =N  u0   3    u04

y1

y2

y3

y4



 

σ102

0 σ12

0 σ13

0 σ14

 0       0   σ 0 σ 02 σ 0 σ 0 2 23 24    12   ,   0   σ 0 σ 0 σ 02 σ 0    13 23 3 34    0 0 0 σ14 σ24 σ34 σ402 0

        

(3.5)

  1 if y ∗ > 0 1 =  0 otherwise   1 if y ∗ > 0 2 =  0 otherwise

  1    = 0     n.a.d.   1    = 0     n.a.d.

if y1∗ > 0 and y3∗ > 0 (any medical treatment in a private hospital) if y1∗ > 0 and y3∗ ≤ 0 (no medical treatment in a private hospital)

otherwise if y3∗ > 0 and y4∗ > 0 (only an “invasive” treatment in a private hospital) if y3∗ > 0 and y4∗ ≤ 0 (no “invasive” treatment in a private hospital) otherwise

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since y3∗ and y4∗ are not appropriately defined if a patient has not received medical treatment from a private hospital. Note that the first two equations say that the intentions about y1 and y2 are simultaneously determined by the exogenous variables in x1 and x2 . This formulation is supposed to deal with the AS issue. To simplify our notation, it is convenient to denote by x the vector formed by all components of the vectors x1 , x2 , x3 and x4 , and write x′i = x′ Sbi

i = 1, 2, 3, 4.

(3.6)

where Sbi is the K × Ki selection matrix, selecting the components of xi from those of x. Also, we define the four-dimensional vectors y ∗ and y as y ∗′ = (y1∗ , y2∗ , y3∗ , y4∗ ), and y ′ = (y1 , y2 , y3 , y4 ) respectively. The simultaneous four-equations model in the equations (3.1) through (3.4) is a model with a mixed structure in the terminology, as suggested by Maddala (1983). It contains the endogenous latent variables (y ∗′ ), as well as an endogenous dummy variable (y2 ), and may be compactly written as

A0 y ∗ = B 0 x + Γ 0 y + u 0 ,

(3.7)

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where 

A0

B0

Γ0

0 0 0  1   −α0 1 0 0 21  =   0 0 1 0   0 0 0 1   ′ 0′  β1 Sb1     β 0′ S ′   2 b2  =    β 0′ S ′   3 b3    ′ β40′ Sb4   0 0 0 0      0 0 0 0    =    0 1 0 0      0 1 0 0



    ,   

(3.8)

3.1. Logical consistency. We observe that the matrix A0−1 Γ0 given by    0 0 0 0     0 0 0 0    0−1 0 A Γ =   0 1 0 0      0 0 0 0 is lower-triangular, with zeros on the diagonal. If there were no selection in the model under consideration, then, applying a result of Schmidt (1982), we could conclude that the model (3.7) is logically consistent without the additional conditions. According to Gourieroux-Laffont-Monfort (1980), this implies that the model has a well-defined reduced form, which expresses the latent variables in terms of the exogenous variables and the disturbance terms. However, given that the model under consideration is with two endogenous sampling selections, the result by Schmidt (1982) does not seem to be directly applicable. To examine the logical consistency of the model, one could adopt the procedure developed in Gourieroux-Laffont-Monfort (1980). However, a more direct method is in following the approach

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presented in Maddala (1983), that considers under which conditions on the parameters the sum of the different probabilities of the endogenous dichotomous variables is equal to one. To this end, we first observe that, by using (3.7) and inverting A0 , the latent variables can be expressed in terms of y2 and the exogenous variables throught the following system of equations : y ∗ = A0−1 B 0 x + A0−1 Γ0 y + v 0 ,

(3.9)

where v 0 = A0−1 u0 and !  !  D v 0 = N 0, Ω0 , where Ω0 = A0−1 Σ0 A0−1′

(3.10)

and 

02 0  σ1 σ12   σ 0 σ 02 2  12 Σ0 =   σ0 σ0  13 23  0 0 σ14 σ24

Similarly, Ω0 may  02  ω1   ω0  12 0 Ω =  ω0  13  0 ω14

 0 0 σ13 σ14   0 0  σ23 σ24   0  σ302 σ34   0 σ34 σ402

(3.11)

be written as 0 ω12

0 ω13

0 ω14

ω202

0 ω23

0 ω24

0 0 ω23 ω302 ω34 0 0 ω24 ω34 ω402

        

Following Heckman (1978), we shall refer to model (3.9) as semi-reduced form model.

(3.12)

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It is useful to write equations (3.9) in detail as follows: y1∗ = x′1 β10 + v10

(3.13)

0 + v20 y2∗ = x′2 β20 + x′1 β10 α21

(3.14)

0 y3∗ = x′3 β30 + γ32 y2 + v30

(3.15)

0 y4∗ = x′4 β40 + γ42 y2 + v40

(3.16)

We now turn to the expression of different probabilities of the endogenous dichotomous variables. To this end, we first observe that due to the endogenous sampling selections, the probabilities P {y1 = 0, y3 , y4 } are identically equal to zero, for any values of (y3 , y4 ). Next, in order to derive the remaining probabilities, we define Pij0

= P 0 (y1 = i, y2 = j)

(i = 0, 1; j = 0, 1)

0 Pijk = P 0 (y1 = i, y2 = j, y3 = k)

(i = 0, 1; j = 0, 1; k = 0, 1)

0 = P 0 (y1 = i, y2 = j, y3 = k, y4 = m) Pijkm

(i = 0, 1; j = 0, 1; k = 0, 1; m = 0, 1)

The eight different possible sets of values for y1 , y2 , y3 and y4 , together with their respective probabilities, are given in the following table:

y 1 y2 y3 y4

P robability

0

0

0 P00

0

1

0 P01

1

0

0

1

0

1

0

0 P1010

1

0

1

1

0 P1011

1

1

0

1

1

1

0

0 P1110

1

1

1

1

0 P1111

0 P100

0 P110

 = P v10  = P v10  = P v10  = P v10  = P v10  = P v10  = P v10  = P v10

0 ) < −x′1 β10 , v20 < −(x′2 β20 + x′1 β10 α21 0 ) < −x′1 β10 , v20 ≧ −(x′2 β20 + x′1 β10 α21



0 ), v 0 < −x′ β 0 ≧ −x′1 β10 , v20 < −(x′2 β20 + x′1 β10 α21 3 3 3



0 ), v 0 ≧ −x′ β 0 , v 0 < −x′ β 0 ≧ −x′1 β10 , v20 < −(x′2 β20 + x′1 β10 α21 4 4 3 3 4 3 0 ), v 0 ≧ −x′ β 0 , v 0 ≧ −x′ β 0 ≧ −x′1 β10 , v20 < −(x′2 β20 + x′1 β10 α21 3 3 3 4 4 4 0 0 ′ 0 0 0 ′ 0 ′ 0 0 ′ ≧ −x1 β1 , v2 ≧ −(x2 β2 + x1 β1 α21 ), v3 < −(x3 β3 + γ32 )



0 ), v 0 ≧ −(x′ β 0 + γ 0 ), v 0 < −(x′ β 0 + γ 0 ) ≧ −x′1 β10 , v20 ≧ −(x′2 β20 + x′1 β10 α21 42 4 4 4 32 3 3 3 0 ), v 0 ≧ −(x′ β 0 + γ 0 ), v 0 ≧ −(x′ β 0 + γ 0 ) ≧ −x′1 β10 , v20 ≧ −(x′2 β20 + x′1 β10 α21 3 3 3 32 4 4 4 42



(3.17)

16

It is easy to verify that 0 0 P00 + P01 = P {v10 < −x′1 β10 }

(3.18)

0 0 0 ), v30 ≥ −x′3 β30 } P1010 + P1011 = P {v10 ≥ −x′1 β10 , v20 < −(x′2 β20 + x′1 β10 α21

(3.19)

0 0 0 0 ), v30 ≥ −(x′3 β30 + γ32 )}, P1110 + P1111 = P {v10 ≥ −x′1 β10 , v20 ≥ −(x′2 β20 + x′1 β10 α21

(3.20)

and thus 0 0 0 0 )}. P110 + P1110 + P1111 = P {v10 ≥ −x′1 β10 , v20 ≥ −(x′2 β20 + x′1 β10 α21

(3.21)

Also, we have 0 0 0 0 P100 + P1010 + P1011 = P {v10 ≥ −x′1 β10 , v20 < −(x′2 β20 + x′1 β10 α21 )}.

(3.22)

Therefore, we get from (3.21) and (3.22) that 0 0 0 0 0 0 P100 + P110 + P1010 + P1011 + P1110 + P1111 = P {v10 ≥ −x′1 β10 },

(3.23)

and, finally, using (3.18), 0 0 0 0 0 0 0 0 P00 + P01 + P100 + P110 + P1010 + P1011 + P1110 + P1111 =1

(3.24)

Equation (3.24) implies that the sum of different probabilities of the endogenous dichotomous variables is equal to one and, therefore, the model (3.7) is logically consistent without imposing any additional restrictions on the parameters. Once we have shown the existence of the reduced form for the latent variables, in terms of the exogenous variables and the disturbance terms, where the next step consists in considering the identification of the model. To this end, below, we explicit the constraints imposed on the parameters of the model.

3.2. Constraints on the parameters. In addition to the constraints on the matrices A0 , B 0 and Γ0 implied by the exclusion restrictions, there are a number of restrictions resulting from the specification of the model.

17

To see this, it is useful to write equation (3.10) in details as follows: 

ω102

0 ω12

0 ω13

0 ω14

   ω 0 ω 02 ω 0 ω 0 2 23 24  12   ω 0 ω 0 ω 02 ω 0  13 23 3 34  0 0 02 ω14 ω24 ω34 ω40





σ102

0 σ 02 α21 1

0 σ12

0 σ13

0 σ14

+       α0 σ 02 + σ 0 α02 σ 02 + 2α0 σ 0 + σ 02 α0 σ 0 + σ 0 α0 σ 0 + σ 0 12 21 1 21 12 2 21 13 23 21 14 24   21 1 =   0 0 0 0 02 0 σ13 α21 σ13 + σ23 σ3 σ34     0 0 σ0 + σ0 0 σ14 α21 σ34 σ402 14 24 (3.25)

As y1∗ , y2∗ , y3∗ and y4∗ are observed as dichotomous variables, we need to impose the following conditions ω102 = 1;

ω202 = 1;

ω302 = 1;

ω402 = 1

Therefore, the first and second order structural form parameters are subject to the additional constraints: σ102 = 1

(3.26)

02 0 0 α21 + 2α21 σ12 + σ202 = 1

(3.27)

σ302 = 1

(3.28)

σ402 = 1

(3.29)

such that,



0 σ12

0 σ13

0 σ14

 1   σ 0 σ 02 σ 0 σ 0 2 23 24  12 0 Σ =  σ0 σ0 0 1 σ34  13 23  0 0 0 σ14 σ24 σ34 1

        

0 and σ 02 are related to α0 according to (3.27), and where σ12 2 21

(3.30)

        

18



ω102

0 ω12

0 ω13

0 ω14

   ω 0 ω 02 ω 0 ω 0 2 23 24  12 0 Ω =  ω 0 ω 0 ω 02 ω 0  13 23 3 34  0 0 0 ω14 ω24 ω34 ω402





0 α21

0 σ12

0 σ13

0 σ14

1 +     0 σ0 + σ0 0 0 0   α0 + σ 0 1 α21 12 13 23 α21 σ14 + σ24   21 =   0 0 σ0 + σ0 0 σ13 α21 1 σ34   13 23   0 0 σ0 + σ0 0 σ14 α21 σ34 1 14 24 (3.31)

There is also another set of restrictions related to the assumptions that the matrices Σ0 and Ω0 are positive definite. This is equivalent to the requirement that the determinants associated with all upper-left sub-matrices of Σ0 and Ω0 are positive. This implies that the determinants of these matrices are positive, so, they are nonsingular. In many situations, the unconstrained estimates of variance-covariance matrices of nonlinear econometric models satisfy this set of constraints. However, in the model, there is nothing to guarantee that it will be the case. Therefore, in principle, we would need to apply a numerical estimation procedure, which imposes a set of restrictions guaranteeing that the estimates of the matrices Σ0 and Ω0 are positive definite. However, given the difficulties in implementing such a numerical procedure for the estimation of our model, we have adopted an alternative strategy, which will be described in Subsection 3.4.

3.3. Identification of the full model. In order to estimate the model structural parameters, we need to guarantee the identification of the model, which has to be checked once the logical consistency conditions are satisfied. The identifiability conditions in a mixed structure model, like our model, are somewhat different from those in the standard simultaneous-equations models. Following Huguenin (2004), we shall examine the identifiability of the full model, by applying the well-known concept of equivalent structures, as first stated in Koopmans (1949) and developed in Koopmans et al. (1950). Assuming normality of the distribution of the vector u0 , a structure S 0 consists of a set of values of the coefficient matrices A0 , B 0 , Γ0 and the variance-covariance matrix Σ0 , and can be conveniently written as S 0 = (A0 , B 0 , Γ0 , Σ0 ).



    ,   

19

Let M be a nonsingular matrix and consider the linear transformation of model (3.7), M A0 y ∗ = M B 0 x + M Γ0 y + M u0 , !  !  D M u0 = N 0, M Σ0 M ′

(3.32) (3.33)

Let S ⊕0 be a structure derived by S 0 by the transformation A⊕0 = M A0 ,

B ⊕0 = M B 0 ,

Γ⊕0 = M Γ0 ,

Σ⊕0 = M Σ0 M ′

(3.34)

Clearly, the structures S 0 and S ⊕0 = (A⊕0 , B ⊕0 , Γ⊕0 , Σ⊕0 ) share the same semi-reduced form. Therefore, given that the logical consistency conditions are satisfied, they necessarily share the same reduced form, which expresses the latent variables, in terms of the exogenous variables and disturbance terms. For this reason, S 0 and S ⊕0 are (observationally) equivalent. We now consider the set of structures S ⊕0 which satisfy all the a priori restrictions on A0 , B 0 , Γ0 and Σ0 . We shall say that the structure S 0 is identifiable by this set of a priori restrictions if there is a unique equivalent structure S ⊕0 which satisfies the same set of a priori restrictions. Obviously, this uniqueness property implies that there is a unique matrix M , which produces an equivalent structure, namely the matrix I4 , which is an identity matrix of order four. Direct verification of the identification of model (3.7) is quite easy to carry-out, given the simple triangular structure and the small number of equations of the model under consideration. Identifiability conditions of the equations can be applied to the full model, including some secondorder restrictions on the covariances of the disturbance terms. As shown in Appendix B, under 0 = 0, the system of equation (3.7) is identified without introducing further the assumption that σ12

restrictions on the model. It that case, the matrices Σ0  0 0 σ13  1   0 σ 02 σ 0 2 23  Σ0 =   σ0 σ0 1  13 23  0 0 0 σ14 σ24 σ34

and Ω0 given in (3.30) and (3.31), respectively, are equal to  0 σ14   0  σ24  (3.35) . 0  σ34   1

20



ω102

0 ω12

0 ω13

0 ω14

   ω 0 ω 02 ω 0 ω 0 2 23 24  12 0 Ω =  ω 0 ω 0 ω 02 ω 0  13 23 3 34  0 0 0 ω14 ω24 ω34 ω402





0 α21

0 σ13

0 σ14

  1   0 σ0 + σ0 0 0 0   α0 1 α21 13 23 α21 σ14 + σ24   21 =   σ 0 α0 σ 0 + σ 0 0 1 σ34   13 21 13 23   0 0 σ0 + σ0 0 σ14 α21 σ34 1 14 24

        

(3.36)

3.4. The FIML function and numerical computation issues. Before discussing the estimation procedure, we present a simplified specification of our model. Indeed, the matrices Σ0 in (3.35) and Ω0 in (??) could be further simplified if we consider that our main interest with this work is to understand the relationship between the presence of health insurance and the consumption of healthcare services. Therefore, to avoid introducing noisy and irrelevant information in the model, which could affect the estimate of the relevant parameters, we should leave unconstrained only the covariances that link the “insurance choice” (eq. 3.2) with the “consumption 0 = σ 0 = σ 0 = σ 0 = 0. These choice” (eq. 3.3 and eq. 3.4). Implicitly, this leads to set σ13 14 31 41

hypotheses can be easily tested through a likelihood ratio test, as we did. Finally, given the structure of our model and its economic interpretation, we need to set 0 = 0. The rationale for imposing this constraint comes from considering that equations 3.3 σ34

and 3.4 are used to disentangle the “demand side” MH from SID by direct comparison. As will be better discussed in section 3.5, equation 3.3 allows estimating the overall effect of “demand side” MH and SID, while equation 3.4 will estimate only the SID effect. “Demand side” MH will be obtained as a difference. As such, y4 is a subsample (by selection) of y3 . If we link these two equations, then, one of the two equations can absorb the effect of the other, especially if we are in a situation where the overall effect if driven mainly or exclusively by SID and no “Demand side” 0 it MH is present. In this specific context, if we link the two equations through the parameter σ34

21

is equivalent to impose multicollinearity across equations, thus leading to numerical convergence problems as the determinant of Ω0 becomes negative and, therefore, Σ0 cannot be estimated.5 If these restrictions hold, it then follows that the Ω0 matrix could be conveniently rewritten as:



ω102

0 ω12

0 ω13

0 ω14

   ω 0 ω 02 ω 0 ω 0 2 23 24  12 Ω0 =   ω 0 ω 0 ω 02 ω 0  13 23 3 34  0 0 0 ω14 ω24 ω34 ω402





0 α21

0 0   1   0 0   α0   21 1 σ23 σ24 =   0 σ0 1 0   23   0 0 σ24 0 1

        

(3.37)

Estimation of the model (3.7) by full information maximum likelihood (FIML) method can be carried out based on results presented earlier. The FIML function can be expressed in terms of the eight different possible sets of values for y1 , y2 , y3 and y4 , together with their respective probabilities given in Table (3.17). Specifically, taking into account the two endogenous sample selections present in the model, the Log-likelihood function for the complete model can be written compactly as:

LN =

1 1 X N X X

n=1 y2 =0 y4 =0

ln [(1 − yn1 ) Pn0y2 + yn1 (1 − yn3 ) Pn1y2 0 + yn1 yn3 Pn1y2 1y4 ]

(3.38)

where, from Table (3.17),

Pn0y2

  P {vn1 < −x′ β1 , vn2 < −(x′ β2 + x′ β1 α21 )} if y2 = 0 n1 n2 n1 = ,  P {v < −x′ β , v ≥ −(x′ β + x′ β α )} if y = 1 n1 2 n1 1 n2 n2 2 n1 1 21

5Based on our data, we can show that the determinant of the Ω0 matrix becomes positive for values of σ 0 above 34 0 -0.905, while the point estimate of our σ34 is -0.962. For a justification of this result see Appendix C. Clearly, these problems could be solved if we were able to implement a numerical estimation procedure which imposes that the estimates of the matrices Σ0 and Ω0 are positive definite. In this way we could further obtain an estimate 0 of the matrix Σ0 and then test for the restriction σ34 = 0. We are perfectly aware of this problem and that the solution we proposed, although reasonable from both an economic and econometric point of view, is not the most satisfactory, thus representing a potential limitation of our work. However, given the complexity of the problem at stake in terms of redefining completely the computational aspects of the optimisation process, we leave this for future work in our research agenda.

22

Pn1y2 0

  P {vn1 ≥ −x′ β1 , vn2 < −(x′ β2 + x′ β1 α21 ), vn3 < −x′ β3 } if y2 = 0 n1 n2 n1 n3 = ,  P {v ≥ −x′ β , v ≥ −(x′ β + x′ β α ), v < −(x′ β + γ )} if y = 1 n1 1 n2 2 1 21 n3 3 32 2 n1 n2 n1 n3

Pn1y2 1y4

  P {vn1       P {vn1 =   P {vn1      P {v n1

≥ −x′n1 β1 , vn2 < −(x′n2 β2 + x′n1 β1 α21 ), vn3 ≥ −x′n3 β3 , vn4 < −x′n4 β4 }

if y2 = 0 and y4 = 0

≥ −x′n1 β1 , vn2 < −(x′n2 β2 + x′n1 β1 α21 ), vn3 ≥ −x′n3 β3 , vn4 ≥ −x′n4 β4 }

if y2 = 0 and y4 = 1

≥ −x′n1 β1 , vn2 ≥ −(x′n2 β2 + x′n1 β1 α21 ), vn3 ≥ −(x′n3 β3 + γ32 ), vn4 < −(x′n4 β4 + γ42 )}

if y2 = 1 and y4 = 0

≥ −x′n1 β1 , vn2 ≥ −(x′n2 β2 + x′n1 β1 α21 ), vn3 ≥ −(x′n3 β3 + γ32 ), vn4 ≥ −(x′n4 β4 + γ42 )}

if y2 = 1 and y4 = 1

Computation of these probabilities requires integration of the multivariate normal density over infinite range subsets of R2 , R3 and R4 . Extending the work of Lazard-Holly and Holly (2002), Huguenin (2004) showed how to transform multiple infinite range integrals of the multivariate normal density into a sum of lesser order multiple finite range integrals, where the domains of integration are bounded and delimited by the correlation coefficients.6 A further advantage of this analytical decomposition is to allow for the expression of the gradient of the log-likelihood function with respect to any of its parameters in a simple way. The BHHH approach (see Berndt et al., 1974) lends itself naturally well in this setting, as it avoids computing the analytical Hessian function of the log-likelihood. Numerically, both the log-likelihood function and the gradient function involve small finiterange multiple integrals that can be evaluated by Gauss-Legendre quadrature, and the maximisation can be performed with the help of a numerical Hessian. Despite the presence of some routines that estimate multivariate Probit using simulation methods, all estimates in this paper have been obtained adapting to the quadrivariate case the numerical exact methods and estimation techniques programmed in GAUSS for the trivariate case by Huguenin (2004), using the maximum likelihood library MAXLIK. According to Huguenin (2004) and Huguenin et al. (2014), the estimates obtained with this FIML procedure seem to be more accurate than those based on SML methods. We also believe that these routines represent an improvement with respect to existing software for at least a couple of reasons.

6See also Huguenin et al. (2014) for an exposition of the main results of Huguenin (2004).

23

In particular, Roodman (2011, 2014) proposed a routine (cmp Stata command), which gives the possibility to estimate a class of models like the one proposed in this paper using Simulated Maximum Likelihood (SML) techniques. Roodman’s procedure is based on the Geweke-HajivassiliouKeane (GHK) algorithm, to compute higher dimensional cumulative normal distributions. As highlighted by Roodman (2014) and by Cappellari and Jenkins (2006), results obtained using procedures based on the GHK simulator may differ depending on the sort order of the data, given that the sort order affects which values of the random variable(s) get allocated to which observation7. According to Train (2009), this problem could determine two kind of distortion, classified as simulation noise and simulation bias, which could be attenuated by simply increasing the number of random draws used for the simulation. In fact, if the number of random draws rises faster than √ N , these problems should be easily solved. In order to check this fact, we run the cmp command using 4 different sort-order datasets and compare how the simulation noise and simulation bias evolve by number of replications (from 50 to 1,000). Therefore, for each parameter, we have four different estimates, from which we derive a Standardised Dispersion Index (SDI); in order to give the same information at equation level, we compute an average which is based on (J) values, where J is the number of parameters in the equation.8 In table 1, we show the results for the simulation noise. For the first two equations, the noise seems to be very low and disappears rapidly even in the presence of a small number of draws. On the contrary, the noise is higher for the last two equations related to the endogenous sample selection procedure and it persists even when the number of draws is triple with respect √ to what is suggested in the literature ( N ≈ 323). Additionally, this noise is higher when we look at our parameters of interest. Concerning the simulation bias, we run a similar experiment, by comparing the average estimate obtained from the 4 different order datasets using the cmp command to the estimate obtained 7Differently from our routine, the cmp procedure doesn’t attain convergence when we cosider σ 0 6= 0 and σ 0 6= 0. 13 14

Therefore, the comparison p is based on Equation ??. V ar(αi ) 8In particular, SDI = , where αi is the parameter of interest and i represents different cmp estimates j α¯i PJ 1 SDIj and eq.n ¯ = , where j is the number of parameters in each equation. J

24

using our FIML estimator, which should not be affected by this type of bias, since we do have the exact solution procedure. In table 2, we show for each equation and for the parameters of interest where the mean of the absolute bias is in the percentage point. As before, we can see that for the first two equations, the bias seems to disappear quickly when we increase the number of draws. However, the bias is higher and more persistent for the last two equations, which involve double endogenous sample selection process. Finally, comparing our results with the simulated ones, our estimation is also more efficient: about 70% of the estimated parameters have a lower standard error with respect to the simulated one, whatever is the number of draws used. These results confirm that the FIML procedure we use gives more accurate results. In fact, in our specific case, both simulation noise and simulation bias, at least for the equations that involve the double selection, seem to be more persistent than how it should be according to the literature. 3.5. Testing and disentangling AS, MH and SID: the identification strategy. Given this econometric framework, testing for AS is a rather simple task, while measuring and disentangling MH and SID remains more complicated. The presence of adverse selection is related with the endogeneity of y1∗ in Eq.(3.2). Under the 0 + ρ0 = 0. normality assumption of our model, y1∗ is exogenous in Eq.(3.2), if and only if α21 12

Given that the covariance between the perturbations of Eq.(3.1) and Eq.(3.2), (ρ012 ), has been set to zero for identification reasons, which follows a simple t-test on the significance of the parameter 0 (associated with the variable “Total Healthcare Demand ” in Eq.(3.2)), and that is a test for α21

the presence of AS. Unfortunately, testing for the presence of MH and SID, and disentangling the two effects, is more difficult, as both effects work in the direction of increasing healthcare utilisation. The only chance we have to separate them is to exploit the existence of the potential differences in the way patients can see the demand for healthcare services. Therefore, our identification strategy relies on the possibility to separate healthcare services which are “invasive”, and therefore, supposed to be unwanted or avoidable if possible by the patients, from those who are not. Clearly, our maintained hypothesis is that the “invasive” category should be less affected (or not affected at

25

all) by MH effect, and if an effect is found, should be imputed to SID. On the contrary, for “non invasive” procedures, we could observe MH and SID at the same time. Simply computing the difference between the two parameters should lead to identifying the two effects. This is why, in Eq.(3.3), the variable y3∗ include all healthcare services provided in a private setting (the more sensible to private insurance effect), theoretically affected by both MH and SID, while in Eq.(3.4) the dependent variable is restricted to the subset of “invasive” private healthcare services, which 0 could help us to identify SID. should not be affected by MH. Given this setup, a t-test over γ42

Given the selection structure between Eq.(3.3) and (3.4), and under the assumption that both γ32 and γ34 > 0 (i.e., MH cannot be negative), the following different meaningful alternatives can be obtained when comparing and interpreting the relative importance of “demand side” MH vs. SID: 0 > 0 and γ 0 = 0, implies that we have “demand side” MH but not SID; (1) if γ32 42 0 > γ 0 and both are > 0, implies that SID is lower than “demand side” MH; (2) if γ32 42 0 ≤ γ 0 and both are > 0, implies that SID is greater than “demand side” MH, (3) if γ32 42

although we cannot exclude that “demand side” MH is null (it is observationally equiv0 = γ 0 > 0, which implies that the total alent). This case includes also the sub case γ32 42

MH effect is supply-driven. 0 is not Although potentially available, for obvious reasons, we rule out situations where γ32 0 is statistically significant. Finally, a simple Wald test can be statistically significant, whereas γ42 0 6= γ 0 . used to easily test if γ32 42

Finally, if we can exclude a priori the existence of MH or SID, then, the empirical model can be reduced to a simpler trivariate Probit model. 4. Data used in the empirical analysis The data used for the empirical analysis are from the ISTAT Multiscopo Survey (MHS), conducted by Italian National Institute of Statistics in 1999-2000. This survey is conducted each five years, with the aim to describe and analyze the health conditions and the use of the health care services (GPs, specialists - public and private - and hospital care and related health care services) of the Italian population. A representative sample of households and individuals are interviewed

26

to have a picture of the health conditions and access to healthcare services of the Italian population. Unfortunately the two more recent waves of this survey do not contain information about additional insurance coverage. The used sample for the 1999-2000 survey is made of 52,332 households, with an equivalent of 140,011 individuals. After dropping individuals younger than 18 years of age and observations with missing values, the final sample used in this analysis consists of 104,422 observations.

9

The covariates chosen in our study are typical of those used in previous studies on the determinants of medical care. In particular, the explanatory variables include the variables reporting the health status of the individuals, as well as income, gender, age, education, and geographic location. On top of these, we have also used supply side variables to help control the identification of supply induced demand effects. In Table 3, we report the descriptive statistics about the variables used in the empirical analysis. Concerning the dependent variables, in a given month, about 36 percent of individuals had a control, a treatment or spent at least one day in a hospital. In addition, 13 percent declared to have a supplementary private health insurance and only 20 percent of the population used private healthcare services. Although not reported in the table, it is interesting to note that in Italy the health care system public and private specialist use is roughly the same, in spite of the fact that visits to private specialists are considerably more expensive. Information on health status is provided by three different variables: limitation in daily activities, self-reported health status, and body mass index. With regards to the first variable, the question asked to individuals is “Are you hampered in your daily activity by any physical or mental health problem, illness or disability?” In this case, the dummy variable is based on three categories: “yes, severely”, “yes, to some extent”, and “no”. In the same way, respondents rate their health status by choosing between five categories: “very bad”, “bad”, “fair”, “good” and “very good” health conditions. Finally, the body mass index is constructed as the ratio of weight to the square of height. Two variables describing “under-weight” and “over-weight” people have then been constructed. They do not add to 1 because they are defined as lower or higher than 9Children up to the age of 18 years were not considered for two main reasons. First, their access to healthcare goes mainly through SSN pediatricians, that are considered specialists, but operate as primary care physicians. Second, in general, the decision to access healthcare services for children is mediated by their parents.

27

some medically accepted thresholds (18.5 and 25 respectively). The “normal-weight” body mass index is the reference variable. This allows for the non-linearity of the effect of the body mass index on patient health. The majority of the individuals declared that they were in good health. In fact, 12.0 percent of people considered their health as “very good”, 42.6 percent as “good”, and 37.3 percent as “fair”. Only about 8 percent of the population declared a “poor” or “very poor” health status. People who reported in general to be in a “very poor” or “poor” health status were concentrated in the lower income quintiles. It is also interesting to note that from the geographical perspective, the share of people in either “good” or “very good” health status, which was higher in the North compared to the South. Information on income is not directly available in the our database. However, excellent measures of income are available in the Survey of Household Income and Wealth (SHIW) conducted by Bank of Italy for the year 2000. We then use an imputation procedure, based on the cell matching to impute income data from the Bank of Italy survey in the Multiscopo survey (details are reported in Atella and Pollastri, 2003)10. For the education level, respondents could choose among nine categories, that have been collapsed in four widely accepted categories: “Up to elementary degree”, “Middle school degree”, “High school degree” and “University degree”. More than half of the sample (66.1 percent ) had up to the middle school degree. Only 6.5 percent of the sample held a university degree. The sample is almost equally divided into males (47.0 percent) and females (53.0 percent), with a large fraction living in the North of Italy (40.0 percent) and in the South (42.0 percent). A small fraction of individuals (7.0 percent) report being in poor health, although 51.0 percent report a chronic condition. The average age of the population (after dropping all children below 18 years) under investigation is 47 years. About 10.7 percent of the population is exempted from the payment for illness reasons, while another 10.6 percent is exempted for other reasons (income, unemployment status, invalid, etc.). 10Although we are aware of the fact that the use of imputed income introduces measurement errors, we must also

recognize that the MHS is an incredible source of individual-level information on the demand of healthcare services in Italy that would otherwise be under-utilised. This means also that in commenting the empirical results our conclusions on the income parameter should be very cautious. In any case, Atella and Deb (2004) have shown that the introduction of imputed income does not alter the magnitude and significance of the other variables.

28

One third of the population, over the last 12 months, did not get any diagnostic control (blood pressure, cholesterol test, pap test, etc.), while about 65.0 percent at least got one control. More than 44.4 percent of the patients in the sample was a smoker during the time of the interview. About 19.3 percent of the households involved in the survey were childless couples, 59.4 percent were couples with children, 9.5 percent singles, while the remaining were mainly single parents households (of which lone mothers - about 7.0 percent - were the majority), and other more complex typologies. Finally, our set of regressors include five “supply-side” variables, collected at the regional level. Supply variables are of particular interest when modelling the healthcare service demanded by patients. Among other things, they should help the identification of the parameters in the empirical analysis. The first two variables are related to the availability of GPs on the territory. We have been able to collect information, at regional level, on GP density (number of GPs per patient) and on the distribution of patients across GPs. In fact, although the distribution of GPs across the Italian region is an interesting phenomenon to capture, the way in which patients are allocated to GP within a region is another important aspect as well. In other words, we can have the same amount of GPs per patients in two regions, but in one region, patients are equally allocated to the GPs available, while, in the other region, some GPs could concentrate most of the patients while some other have very few patients. This difference may cause serious problems, in terms of gate-keeping and referral, thus, causing pressure problems for healthcare demand. The remaining three variables are specialist density (number of specialists per Km2 ), the number of bed per patient in public hospitals and in private hospitals. In the Italian public system, there is no co-payment to pay for hospitalization in public structures. Co-payments are due only in case the patients require a better accommodation (single room with more comfort). However, this kind of accommodation is limited by the fact that many hospitals do not have adequate infrastructure to offer such services. A completely different scenario exists for private hospitals. In this case, the moral hazard effects may be extremely high. This also explains why, in the empirical model, we have used private utilization of healthcare as dependent variable for one of our equations.

29

With respect to the adverse selection effect, we know that it is more relevant in that part of the population that may incur a higher probability of using hospitalization. However, the effect may depend also on the relative availability of public and private hospital structures. In those regions where public hospitals work properly and efficiently, the adverse selection effect may be lower. On the contrary, where the public hospitals are not so efficient and of low quality the effect may be greater. In any case, private insurers can refuse the renewal of the insurance plan to those patients with high expenditure records or with specific conditions. This is why, in the econometric model, it is important to provide some conditions on the regional differences and supply variables, such as the number of GPs and public specialist per patient. Tables 4 and 5 present some statistics concerning the utilization of services by the patients according to their gender, age, classes and the presence of a private insurance plan. Table 6 shows the number of insurance plans and types of health service utilization, according to age and sex, while table 7 shows the proportions of the patients with private health insurance, together with the proportion of patients utilizing health services. A quite interesting result emerges from the analysis of these data. In fact, while the proportion of patients who use public healthcare services has a common U-shaped form – a minimum at the age of 35-49 – an almost flat pattern comes out from the private healthcare service utilization. In any case, the groups of older patients have a lower utilization of private healthcare services (partially explained by the lower levels of income that characterized pensioners). In table 7, we observe that while the percentage of patients with private insurance accessing public healthcare services is not different from those without it, differences exist between the insurance plans for access to private health services. In fact, the percentage of “private services” is higher for the patients with “private insurance” compared to those relying only on the public insurance plan (SSN). Of course, this result stems from the peculiarity of the Italian system being an SSN.

5. The econometric results In this section, we comment on the econometric results obtained from the estimation of different models. Equations (3.13) and (3.14) have been estimated using a sample containing 104,556

30

observations, while Equation (3.15) has been estimated conditional on the patient’s use of public healthcare services, thus, using only 38,352 observations, and Equation (3.16) conditional on patient’s use of private healthcare services, thus using only 21,440 observations. Overall, the model is well-determined and the parameter estimates are statistically significant, together and with the expected signs. In the few cases in which they are not, they are concentrated in case where the lack of significance is not a problem for our hypotheses. In table 8, we report the model estimates. These results are obtained using a matrix Ω0 0 = like in matrix (3.37), given that the value of the LR test for the hypothesis, where H0 : σ13 0 = σ 0 = σ 0 = 0 is equal to 4.94, suggesting that the null is not rejected at 5% level. The σ14 31 41

empirical results are strongly and neatly in favour of the presence of endogeneity and crossequation error correlation. These results suggest that there is a self-selection problem in the Italian health insurance market, which is in line with the theory and literature (see Holly et al., 1998; Jones et al., 2007; Buchmueller et al., 2004); Dardanoni et al. (2012)). In fact, the positive 0 ) in the second equation shows that a and statistically significant latent variable parameter (α21

higher probability to demand medical treatment is positively related to a higher probability to apply for supplementary private insurance (adverse-selection). Having controlled for AS, the estimates obtained in Equations (3.15) and (3.16) will allow to disentangle the reasons leading privately insured patients to “over-consume” healthcare services (SID vs. MH). Focusing on Equation (3.15), we see that the probability to receive a treatment 0 = from private hospitals is higher if the patients own a supplementary private insurance (γ32

0.2515), which implies the presence of MH. However, what we do not know yet is whether this “over-consumption” behaviour is caused either by the “demand side”’ MH or by SID, or by both. 0 in Equation In order to disentangle these two effects, we need to focus on the parameter γ42

(3.16), where the dependent variable has been restricted to the subset of “invasive” treatments received in private hospitals, which we assume are difficult to be induced by the demand-side 0 associated to the private insurance variable in MH. In table 8, we see that the parameter γ42 0 in Equations (3.15); nevertheless is the value of the Equations (3.16) is twice the parameter γ32

Wald test at the bottom of the third column in table 8 informs us that the difference between the two parameters is not statistically different from zero. This result is equivalent to alternative

31

3 seen in Section 3.5, which implies that SID is more relevant than the “demand induced” MH, and probably, the total MH effect shown in the third equation is supply-driven. As already discussed in Section 3.5, the only way to understand the magnitude of the “demand induced” MH effect is to restrict the type of healthcare services demanded to those services potentially non being affected by SID. In the Italian context, this type could be represented by private specialist visits. In fact, specialist visits are performed by “self-employed” professionals working in independent practices, without strong connections with their colleagues. Within this setting, we could exclude the possibility that one specialist could stimulate the demand for other specialists, particularly in different specialties (i.e., it is quite unrealistic that a psychiatrist will induce demand for an orthopaedist).11 Given this setting, we computed an indicator of demand for health services, which is equal to one, if, in the 4 weeks before the interview, patients have been prescribed more than one outpatient specialist visit in different specialties.12 With this definition of private healthcare demand, we should exclude, by construction, the presence of SID and, therefore, our model reduces to a simpler trivariate Probit. In this new setting, a positive 0 > 0 leads to inferring the existence of “demand induced” and statistically significant value of γ32

MH. However, as we can see from Table 9, the parameter for the presence of additional insurance is slightly positive and not statistically significant. Based on these results, we may conclude against the presence of “demand induced” MH, while in favour of AS and SID. 0 in Equation A final problem that we need to tackle is the low statistical significance of γ42

(3.15), which may casts some doubts about the validity of our hypotheses. This result could be easily explained if we consider that this analysis focuses on privately insured patients, and that the elderly patients can hardly get a private insurance when they are very old (+80). By including these patients in our sample, it is equivalent to dilute the MH effect, given that we include patients who would like to buy an insurance but are refused. Therefore, we have restricted our sample by excluding the patients +80 years of age, who will never have the chance to apply for a private health insurance. The corresponding results are presented in Panel a of Table 10, from which we 11This situation was particularly true in Italy at the time when the survey data we use were collected, as specialists

were not working in practices, and several specialists were working together, which could falsify our claim. Of course, given that we cannot rule out the existence of SID within the same specialist, we exclude them. 12As robustness checks, we have also used different thresholds: “more than two” and “more than three” specialist visits. Results are similar and are available upon request.)

32 0 which now is can see how the estimates remain almost unchanged, except for the parameter γ42

significant at 5%.13 We conclude this section by commenting on the remaining regressors in our model. Concerning socioeconomic determinants, we find that income has a positive and a highly significant effect only in the equation where insurance decision is considered. This is not a surprising result. In fact, in the case of the probability of using public healthcare services, the NHS seems to work properly (this result is supported by evidence obtained by Atella et al., 2004; and Atella and Deb, 2004), while, in the case of private health insurance, a possible explanation could be the existence of collinearity with the educational attainment variable and exemption reasons (in particular, those not linked to health conditions). Across different specifications, age has a non linear effect in equations (1) and (2), while in equation (3), it doesn’t have any effect, except in the specification reported in table 8. Highly educated patients tend to use more healthcare services, and, in particular, to use private services. However, this is not true when we consider invasive treatments. It clearly shows that patients with exemptions tend to use the public system more. At the same time, exempted patients have a lower access to private insurance. Smokers tend to use more both public and private services, as well as more private insurance plans, although this is not true when we consider more invasive treatment provided by private hospitals. Patients who decided to take prevention controls over the previous year have, obviously, a more positive relationship with the dependent variables. In particular, those with high education tend to have both a higher probability of buying a private insurance plan and a higher probability of using private healthcare services. Obviously, this relation is no longer true in the case where we consider more invasive treatments solely. High levels of education are usually associated with high levels of income and, therefore, with high opportunity costs for waiting time for public healthcare services. However, it may also indicate quality effects in the provision of healthcare through private providers. Unfortunately, with the available data, it is not possible to disentangle the 13It is interesting to note that in this case the value of the LR test for the selection of the matrix Ω0 is equal to

2.98 and then is farther from the rejection level. From a “visual check” the estimated parameters do not change, 0 0 6= 0. This seems to confirm that in this 6= 0 and σ14 although the new estimates seem to be less efficient when σ13 case we don’t add information but noise.

33

two effects. Geographical parameters need to be interpreted as differential effects with respect to the reference region, namely Sardegna. While there are some geographical differences for accessing public healthcare structures, we cannot detect any special pattern. However, it is worth mentioning that for two regions, Trentino and Emilia Romagna, considered as the most performing regions for the provision of public healthcare services, we observe positive and strong differential effects. At the same time, for Piemonte, Liguria, Puglia, Campania and Lazio, that are among the less performing, we observe negative values. Finally, the positive and significant effect we observe for Calabria could be explained, considering that this is one of the regions with the lowest level of private structures. A more interesting pattern emerges while analyzing the regional effects on the probability of having a private insurance plan. In fact, in this case, we note that all Northern regions present positive differential effects, with respect to Sardinia, while, almost all Southern regions present negative differential effects. Concerning the third equation, we have not used regional dummies, because our supply side variables have been collected at the regional level. All these variables are significant, and with the expected signs. In particular, as expected, the number of public bed per patient (bed density) have a negative sign on the probability of using private health services. At the same time, the number of specialists per Km2 has a positive effect on the probability to use private services.

6. Robustness check In this section, we perform some robustness checks that allow obtain of more robust and reliable conclusions about our capacity to disentangle AS, MH and SID. Our first check concerns the possibility to distinguish between the patients in terms of discretionary choice. In particular, we know if the choice of a private hospital by a patient is “necessary” or “discretionary”. Intuitively, if the choice is “necessary” we should have less room for SID. For example, if we think of an emergency, it is very likely that the patients choose simply the nearest hospital and the probability to be exposed to SID should be low. Conversely, if the choice is “discretionary”, patients may choose where and when they want to to receive treatments, and, in this case, the possibility to be exposed to some kind of “pressure” in order to undergo some treatment could be higher. Using the previous model, we exploit this information and use it as dependent variable in the third

34

equation a dummy variables that distinguish between “necessary” and “discretionary” choice. As we can see from Table 11, the results seem to confirm our arguments: insured patients seem to consume more healthcare services provided by private hospitals, but only when “discretionary” (Panel c) choices are considered. Additionally, we can distinguish between treatments that should be fully or partially paid by patients (because subsidised by NHS). We expect that the insured should consume more with respect to the uninsured, especially when treatments are fully paid by patients. In fact, in this case, the gain in terms of “demand induced” MH should be higher. On the other hand, if overconsumption by the insured is due because patients are “MH type” we should see if this effect is a form of compensation scheme for the private hospitals. In Panel a and b of table 11, we investigate these features in the data using the previous econometric approach. Empirical results seem to confirm our expectations: the parameter related to insurance is positive and statistically significant, but only when we consider healthcare services not subsidised by the NHS.

7. Conclusions Adverse selection (AS), “demand side” moral hazard (MH) and supply induced demand (SID) are the three most important factors affecting insured patient decisions to use healthcare services. Although there exists a vast theoretical and empirical literature that has approached these phenomena, so far, no contribution has been able to approach them jointly. As largely discussed in the paper, this is due mostly to the difficulty to model the joint determinants of healthcare service utilization and health insurance choice by means of a tractable structural simultaneous equation model. In this paper we have provided a solution to this problem, by presenting a four equation structural model with four latent variables, where the first two equations should ideally deal with the self-selection issue, while the last two equations deal with the over-consumption issues (MH/SID). A closed form solution for the Log-likelihood function has been proposed, and then maximised by the means of FIML using BHHH algorithm. The empirical analysis has been conducted using the Italian cross-sectional data from the 1999 Multiscopo Survey (MS).

35

The empirical analysis has confirmed the theoretical prediction of our structural model. In particular, we found that: 1) the propensity of a person to receive any medical service may not be considered as exogenous in the equation which explains the individual propensity to buy a supplementary health insurance plan; 2) we have not found evidence in favour of “demand ” MH by insured patients. 3) on the contrary, we find that the presence of a supplementary health insurance has a positive effect on demand through SID effect. These results are extremely important from a health policy perspective, given the existing debate on the development of a second pillar in the financing of the healthcare system in Italy and Europe.

36

Appendix A. Tables

Table 1. Simulation Noise

50

100

Number of GHK drawns 200 350 500 700

1000

¯ 0.00 0.00 0.00 0.00 0.00 0.00 eq.1

0.00

¯ 0.00 0.00 0.00 0.00 0.00 0.00 eq.2 α21 0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00

¯ eq.3 0.08 0.07 0.02 0.01 0.01 0.00 γ32 0.26 0.01 0.03 0.06 0.07 0.01

0.00 0.01

¯ 0.35 0.20 0.10 0.12 0.07 0.05 eq.4 γ42 3.68 2.31 1.15 1.16 0.93 0.75

0.05 0.52

Notes. Standardized dispertion index (SDI) within different cmp estimations using 4 different sort order datasets. In case of eq. ¯ the average between the dispersion indices of the all parameters in a specific equation are shown.

Table 2. Simulation Bias Number of GHK drawns 50 100 200 350 500 700

1000

¯ eq.1

0.0

0.0

0.0

0.0

0.0

0.0

0.0

¯ eq.2 α21

0.2 0.0

0.1 0.0

0.0 0.0

0.0 0.0

0.0 0.0

0.0 0.0

0.0 0.0

¯ eq.3 γ32

7.4 4.3

4.6 3.8

0.4 0.1

0.8 0.6

0.6 0.0

0.4 0.0

0.3 0.1

¯ 5.4 eq.4 γ42 16.8

1.8 6.7

1.2 2.1

0.9 1.0

0.8 1.4

0.8 0.2

0.8 0.2

Notes. Absolute bias in percentage point: difference between the average of cmp estimations using 4 different sort order datasets and our FIML estimation. In case of eq., ¯ the average between the percentage bias of the all parameters in a specific equation are shown.

37

Table 3. Descriptive Statistics Variable

Mean Std. Dev.

Min Max

Dependent variables Dummy Dummy Dummy Dummy Dummy

for for for for for

use of health service presence of private insurance use of private health service invasive treatments provided by private hospitals surgery provided by private hospitals Regressors Level of patient’s invalidity Self assessed health status Patient under-weight Patient over-weight Number of prevention tests Patient smoker Patient exempted for pathology Patient exempted for other reasons Household equivalent income Household below poverty line Age of patient / 100 Age squared of patient / 10000 Patient pensioner Marital status Gender Educational attainment Dummy for couples Dummy for couples with children Dummy for single mother Dummy for single father Dummy for singles Number of bed in public hospital per patient Number of bed in private hospital per patient Distribution of patients per GP Number of specialists per Km2

0,367 0,130 0,205 0,006 0,002

0,482 0,337 0,404 0,075 0,050

0 0 0 0 0

1 1 1 1 1

1,107 3,569 0,023 0,420 1,953 0,444 0,107 0,107 3,107 0,239 0,472 0,255 0,137 0,605 1,531 2,107 0,193 0,595 0,067 0,012 0,095 4,609 0,771 0,380 0,236

0,405 1 3 0,841 1 5 0,149 0 1 0,494 0 1 1,800 0 6 0,497 0 1 0,310 0 1 0,309 0 1 1,413 1 5 0,426 0 1 0,181 0,18 0,9 0,183 0,032 0,81 0,344 0 1 0,489 0 1 0,499 1 2 0,950 1 4 0,395 0 1 0,491 0 1 0,251 0 1 0,111 0 1 0,294 0 1 1,397 2,8 10,4 0,471 0 1,9 0,088 0,341 0,786 0,153 0,023 0,546

38

Table 4. Number of insurance plans and type of health service utilization according to age and sex

Age

Patients with only NHS insurance (1)

15-34 35-49 50-64 ≥ 65 Total

29089 20379 21789 19656 90913

15-34 35-49 50-64 ≥ 65 Total

14124 9160 9722 8037 41043

15-34 35-49 50-64 ≥ 65 Total

14965 11219 12067 11619 49870

Patients Patients Patients Patients with any with any with NHS Total with form of form of + Private (1)+(2) hospital health private health insurance visits care service care service (2) utilization utilization Total 4335 33424 959 9442 5257 5281 25660 678 8141 4703 3312 25101 1020 9843 5658 715 20371 1556 10926 5822 13643 104556 4213 38352 21440 Men 2455 16579 333 3934 1985 3097 12257 288 3406 1874 2024 11746 512 4088 2287 408 8445 715 4433 2396 7984 49027 1848 15861 8542 Women 1880 16845 626 5508 3272 2184 13403 390 4735 2829 1288 13355 508 5755 3371 307 11926 841 6493 3426 5659 55529 2365 22491 12898

39

Table 5. Proportion of insurance plans and type of health service utilization according to age and sex

Patients with only Age NHS insurance (1) 15-34 35-49 50-64 ≥ 65

87,0 79,4 86,8 96,5

15-34 35-49 50-64 ≥ 65

85,2 74,7 82,8 95,2

15-34 35-49 50-64 ≥ 65

88,8 83,7 90,4 97,4

Patients Patients Patients Patients with any with any with NHS with form of form of + Private hospital health private health insurance visits care service care service (2) utilization utilization Total 13,0 2,9 28,2 15,7 20,6 2,6 31,7 18,3 13,2 4,1 39,2 22,5 3,5 7,6 53,6 28,6 Men 14,8 2,0 23,7 12,0 25,3 2,3 27,8 15,3 17,2 4,4 34,8 19,5 4,8 8,5 52,5 28,4 Women 11,2 3,7 32,7 19,4 16,3 2,9 35,3 21,1 9,6 3,8 43,1 25,2 2,6 7,1 54,4 28,7

40

Table 6. Number of patients by sex, type of insurance plan, and healthcare service

Patients

Only NHS insurance NHS + Private insurance Total

90913 13643 104556

Only NHS insurance NHS + Private insurance Total

41043 7984 49027

Only NHS insurance NHS + Private insurance Total

49870 5659 55529

Patients with Patients with Patients any form of any form of with hospital healthcare private health visits service care service utilization utilization Total 3817 33390 18464 396 4962 2976 4213 38352 21440 Men 1647 13281 7076 201 2580 1466 1848 15861 8542 Women 2170 20109 11388 195 2382 1510 2365 22491 12898

41

Table 7. Percentage of patients by sex, type of insurance plan, and healthcare service

Patients

Only NHS insurance NHS + Private insurance Total

87,0 13,0 100,0

Only NHS insurance NHS + Private insurance Total

83,7 16,3 100,0

Only NHS insurance NHS + Private insurance Total

89,8 10,2 100,0

Patients with Patients with Patients any form of any form of with hospital healthcare private health visits service care service utilization utilization Total 4,2 36,7 20,3 2,9 36,4 21,8 4,0 36,7 20,5 Men 4,0 32,4 17,2 2,5 32,3 18,4 3,8 32,4 17,4 Women 4,4 40,3 22,8 3,4 42,1 26,7 4,3 40,5 23,2

42

Table 8. FIML estimates of Quadri-variate Probit model

Variables AS, MH and SID parameters Total Healthcare Demand* Dummy for private insurance Supply side variable Density of GPs No. of beds per patient in public hosp. No. of beds per patient in private hosp. Patients per GP Specialist density Other covariates Constant Invalidity status Health status (Good=1) Household equivalent income Dummy for pensioner Age / 100 (Age/100)2 Married Gender (1=male) Educational attainment Dummy for couple Dummy for couple with children Single parent mother Single parent father Household below poverty line Exempted for pathology Exempted for other reasons Patient smoker Number of prevention tests Patient BMI = underweight Patient BMI = overweight Regional dummy (Lombardia as reference) Piemonte Valle d‘Aosta Lombardia Trentino Alto Adige Veneto Friuli Venezia Giulia Liguria Emilia Romagna Toscana Umbria Marche Lazio Abruzzo Molise Campania Puglia Basilicata Calabria Sicilia Macro-regional dummy (Center as reference) North-West North-East South Islands

Total Total Private Private Healthcare Insurance Healthcare Demand Demand

Total Private Healthcare Demand

0.0556***

0.847*** 0.1847*** -0.3321*** -0.0095 0.2018*** -2.517*** 1.8181*** 0.0959*** 0.005848 0.0178*** -0.0619*** -0.138*** -0.0698*** -0.0819** -0.0386*** 0.324*** 0.2174*** 0.0727*** 0.1792*** 0.018661 0.0182**

-2.8101*** -0.0761*** 0.0679*** 0.3695*** 0.012618 7.0027*** -8.1897*** 0.2381*** -0.3158*** 0.2048*** -0.2194*** -0.2141*** -0.1156*** -0.1651*** -0.00915 -0.1064*** -0.1815*** 0.0442*** 0.0404*** 0.016104 0.00895

-0.0657*** -0.0907*** 0.030019 0.0797*** 0.0787*** 0.0727*** -0.0641** 0.1321*** 0.0496** -0.0115 0.030254 -0.03232 0.027332 0.011439 -0.0462** -0.0995*** -0.0073 0.068*** 0.006854

0.2351*** 0.3909*** 0.2647*** 0.6463*** 0.2029*** 0.2315*** 0.1744*** 0.2009*** 0.198*** 0.0882** 0.0753** 0.1639*** 0.036721 -0.02267 -0.3593*** -0.2086*** -0.0765** -0.3042*** -0.2969***

σ32 σ42

-0.1028* -0.266*

Log-Likelihood Total Obs.

-124365 104556

0.2515**

0.5222*

0.0006*** -0.0515*** -0.00249 -2.1728*** 0.1034*

0.0019*** 0.037102 0.0859* -8.6127*** 0.041159

0.013709 -0.0512*** 0.064726 0.0686*** 0.294369 -0.5429** -0.00452 0.013961 0.1116*** 0.1314*** 0.0753** -0.02839 0.028185 0.0329** 0.0553*** 0.03185 0.003636 0.0918*** 0.023491 -0.0253*

-0.1112*** -0.179*** 0.046372 0.030612 -0.74415 0.547605 -0.06745 -0.2005*** -0.03058 0.06046 0.03944 0.026965 -0.0218 0.035232 -0.04154 -0.01617 0.004809 0.0481*** -0.01798 -0.0689*

-0.0442** -0.02985 -0.00502 -0.01432

0.078039 0.065063 0.1151* 0.081996

38352

38352 21440

Wald Test H0 : γ42 − γ32 = 0 (Acceptance level ≤ 3.84) 0.7947 Notes. The model is estimated using an exact FIML with a double selection: y4 is observed only when y3 is equal to one, and in turn y3 is observed only when y1 is equal to one. *** p

< 0.01,

** p

< 0.05, * p < 0.1

43

Table 9. FIML estimates of Tri-variate Probit model - MH for specialistic visits OverTotal Private Consumption Healthcare Insurance of Specialistic Demand Visits

Variables AS, MH and SID parameters Total Healthcare Demand* (α21 ) Dummy for private insurance (γ32 ) Supply side variable Density of GPs No. of beds per patient in public hosp. No. of beds per patient in private hosp. Patients per GP Specialist density Other covariates Costant Invalidity status Health status (Good=1) Household equivalent income Dummy for pensioner Age / 100 (Age / 100)2 Married Gender (male=1) Educational attainment Dummy for couple Dummy for couple with children Single parent mother Single parent father Household below poverty line Exempted for pathology Exempted for other reasons Patient smoker Number of prevention tests Patient BMI = underweight Patient BMI = overweight Regional dummy (Lombardia is the reference) Piemonte Valle d’Aosta Lombardia Trentino Alto Adige Veneto Friuli Venezia Giulia Liguria Emilia Romagna Toscana Umbria Marche Lazio Abruzzo Molise Campania Puglia Basilicata Calabria Sicilia Macro-regional dummy (Center is the reference) North-West North-East South Islands corr2&3

0.0557*** 0.244411 -0.00043 0.0613** 0.0649** 0.834325 0.112454 0.8423*** 0.183*** -0.3315*** -0.01035 0.2026*** -2.5148*** 1.8161*** 0.0958*** 0.006369 0.0177*** -0.0616*** -0.1373*** -0.0689*** -0.0796** -0.039*** 0.3234*** 0.2164*** 0.0738*** 0.1789*** 0.018126 0.0182**

-2.8121*** -0.0783*** 0.0675*** 0.3705*** 0.012312 7.0294*** -8.2226*** 0.2382*** -0.3166*** 0.2051*** -0.2157*** -0.2115*** -0.1124*** -0.1628*** -0.009 -0.1077*** -0.18*** 0.0446*** 0.0404*** 0.014185 0.009212

-0.063*** -0.0879*** 0.032485 0.0813*** 0.0817*** 0.0745*** -0.0643** 0.1352*** 0.0495** -0.01011 0.031277 -0.03096 0.029947 0.012913 -0.0461** -0.098*** -0.0049 0.0709*** 0.007278

0.2344*** 0.3872*** 0.2627*** 0.6454*** 0.202*** 0.2282*** 0.1731*** 0.1989*** 0.197*** 0.0872** 0.0745** 0.1615*** 0.032526 -0.0292 -0.3595*** -0.2121*** -0.0772** -0.3057*** -0.2995***

-1.793*** 0.1118*** -0.195*** 0.037229 0.0932** -0.25381 -0.09775 0.031855 0.0732** 0.0748*** 0.035121 0.055926 0.1026** 0.01216 -0.03636 0.0886*** -0.00236 0.0484** 0.0717*** -0.05172 0.009543

-0.03526 0.004216 0.055025 0.1265*** -0.07732

Log-Likelihood -103718 Total Obs. 104422 38218 Notes. The model is estimated using an exact FIML with a single selection: y3 is observed only when y1 is equal to one. *** p

< 0.01,

** p

< 0.05, * p < 0.1

44

Table 10. Comparison of relevant estimates across different model specifications using a Quadri-variate Probit model Total ♣ Total Private Private Private Healthcare Insurance Healthcare Healthcare Demand Demand Demand

Variables

♣ Total services; limiting patients’ age to ≤ 80

(a)

AS, MH and SID parameters Total Healthcare Demand* (α21 ) Dummy for private insurance (γ32 and γ42 ) Wald Test H0 : γ42 − γ32 = 0 (Acceptance level ≤ 0.004) (b)

< 0.01,

** p

0.573**

0.2495** 0.1892

0.7183*

0.0556***

♣ Surgery interventions; limiting patients’ age to ≤ 80

AS, MH and SID parameters Total Healthcare Demand* (α21 ) Dummy for private insurance (γ32 and γ42 ) Wald Test H0 : γ42 − γ32 = 0 (Acceptance level ≤ 0.004) *** p

0.278** 0.2044

♣ Surgery interventions

AS, MH and SID parameters Total Healthcare Demand* (α21 ) Dummy for private insurance (γ32 and γ42 ) Wald Test H0 : γ42 − γ32 = 0 (Acceptance level ≤ 0.004) (b)

0.0558***

< 0.05, * p < 0.1

0.0558*** 0.2812** 0.3252

0.7092*

45

Table 11. FIML estimates of Tri-variate Probit model - Robustness Check Total Private Healthcare Insurance Demand

Variables

♠ Fully

a AS, MH and SID parameters Total Healthcare Demand* (α21 ) Dummy for private insurance (γ32 )

0.0562*** 0.477***

AS, MH and SID parameters Total Healthcare Demand* (α21 ) Dummy for private insurance (γ32 )

-0.12996

AS, MH and SID parameters Total Healthcare Demand* (α21 ) Dummy for private insurance (γ32 )

AS, MH and SID parameters Total Healthcare Demand* (α21 ) Dummy for private insurance (γ32 ) < 0.01,

** p

< 0.05,

*p

< 0.1

choice of Private hospitals

0.0557*** 0.2668** ♠ Necessary

d

subsidized by NHS

0.0559***

♠ Discretionary

c

*** p

paid by patients

♠ Partially

b

Total Private Healthcare Demand♠

choice of Private hospitals 0.0557*** 0.058092

46

Appendix B. Identification conditions In this Appendix, we derive identification conditions of the system of equations (3.7). Consider a nonsingular matrix   m11 m12 m13   m21 m22 m23  M =  m  31 m32 m33  m41 m42 m43

M given by  m14   m24    m34    m44

We have 

A⊕0

0 m12 α21

 m11 −   m21 − m22 α0 21  0 = MA =   m − m α0 32 21  31  0 m41 − m42 α21



m12 m13 m14   m22 m23 m24    m32 m33 m34    m42 m43 m44

For the matrix A⊕0 to satisfy the same restrictions as A0 given by (3.8), we need to impose the following restrictions on the components of M : all its diagonal components are equal to 1 and all remaining components except possibly m21 are equal to 0. In other words, permissible  0 0 0  1   m21 1 0 0  M =  0 0 1 0   0 0 0 1

matrices M are of the form         

(B.1)

and thus,



A⊕0

1    m21 − α0 21  =  0   0



0 0 0   1 0 0   . 0 1 0    0 0 1

47

Similarly, we have  B ⊕0

   m21 β 0′ S ′ + β 0′ S ′ 2 b2 1 b1  = ′  β30′ Sb3   ′ β40′ Sb4 

Γ⊕0

and

 0 0 0   0 0 0  =  0 1 0   0 1 0



Σ⊕0



β10′ Sb1



    ,   



0   0   , 0    0

0 σ12

0 σ13

0 σ14

1 m21 +    m21 + σ 0 m21 (m21 + σ 0 ) + m21 σ 0 + σ 02 m21 σ 0 + σ 0 m21 σ 0 + σ 0 12 12 12 2 13 23 14 24  =  0 0 + σ0 0 σ13 m21 σ13 1 σ34  23  0 0 + σ0 0 σ14 m21 σ14 σ34 1 24



    .   

Without introducing additional information, it follows that the second equation is not identified. However, if we   1   0  Σ0 =   σ0  13  0 σ14

0 = 0, then Σ0 given by (3.30) becomes assume a priori that σ12  0 0 0 σ13 σ14   0 0  σ202 σ23 σ24  . 0 0 σ23 1 σ34    0 0 σ24 σ34 1

Under this assumption, Σ⊕0 is equal to  0 0 m21 σ13 σ14  1  0 + σ0 0 0  m21 m221 + σ202 m21 σ13 23 m21 σ14 + σ24  ⊕0 Σ =  σ0 m σ0 + σ0 0 1 σ34 21 13  13 23  0 0 + σ0 0 σ14 m21 σ14 σ34 1 24



    .   

(B.2)

48

For Σ⊕0 to satisfy the same restrictions as Σ0 given by (B.2), we need to impose the additional 0 = 0, the system of equation (3.7) restriction m21 = 0. Therefore, under the assumption that σ12

is identified without introducing any further restrictions on the model. In that case, the matrix Ω0 given in (3.31) is equal to



ω102

0 ω12

0 ω13

0 ω14

   ω 0 ω 02 ω 0 ω 0 2 23 24  12 0 Ω =  ω 0 ω 0 ω 02 ω 0  13 23 3 34  0 0 0 ω14 ω24 ω34 ω402





0 α21

0 σ13

0 σ14

  1   0 σ0 + σ0 0 0 0   α0 1 α21 13 23 α21 σ14 + σ24   21 =   σ 0 α0 σ 0 + σ 0 0 1 σ34   13 23 21 13   0 0 σ0 + σ0 0 σ14 α21 σ34 1 14 24

        

(B.3)

49

Appendix C. Invertibility condition for the matrices Σ0 and Ω0 0 the matrices Σ0 and Ω0 are nonIn this Appendix, we consider under which condition on σ34 0 and further assume that σ 0 = 0 and σ 0 = 0. singular, especially when we impose σ12 13 14

In fact, since Ω0 = A0−1 Σ0 A0−1′

(C.1)

where 

A0−1

 1   α0  21 =  0   0



0 0 0   1 0 0    0 1 0    0 0 1

we are only interested in the condition under which Σ0 is non-singular, as it also implies that Ω0 is non-singular since A0 is non-singular with det A0 = 1. 0 = 0., then we have If we further assume for identification purpose that σ12   0 0 0 σ13 σ14   1    0 σ 02 σ 0 σ 0  2 23 24   Σ0 =    σ0 σ0 0  1 σ  13 23 34    0 0 0 σ14 σ24 σ34 1

If we partition Σ0 as 

Σ0 = 

Σ011 Σ012 Σ021 Σ022



,

(C.2)

50

with 



1

0

0

σ202



0 σ13

0 σ14

0 σ23

0 σ24

Σ011 =  Σ012 = 

, 

,

Σ021 = Σ0′ 12 ,   0 1 σ 34  Σ022 =  , 0 σ34 1 we have 0 det Σ0 = det(Σ011 ) det(Σ022 − Σ021 Σ0−1 11 Σ12 )

where det(Σ011 ) = σ202 . 0−1 0 In addition, since Σ011 is non-singular, then Σ0 is non-singular if and only if Σ022 − Σ021 Σ11 Σ12

is non-singular. In details, we have 

0  Σ022 − Σ021 Σ0−1 11 Σ12 =



= 

1

0 σ34

0 σ34

1





−



0 0 σ14 σ24

0 )2 (σ23 (σ20 )2 0 0 0 σ 0 − σ23 σ24 σ13 0 14 (σ2 )2

0 )2 − 1 − (σ13 0 σ34

0 0 σ13 σ23

 

1

0

0

1 (σ20 )2

0 − σ0 σ0 − σ34 13 14

1−

0 )2 (σ24 (σ20 )2



0 0 σ13 σ14



0 σ0 σ23 24 (σ20 )2

0 0 σ23 σ24 



and det(Σ022



0 Σ021 Σ0−1 11 Σ12 )

  0 )2 (σ24 1− = 1− (σ20 )2  2 0 σ0 σ23 0 0 0 24 − σ34 − σ13 σ14 − (σ20 )2 

0 2 (σ13 )

(σ 0 )2 − 23 (σ20 )2

  (C.3)

51 0 or σ 0 or both are equal to zero. Then Assume, for argument’s sake, that σ13 14

det(Σ022



0 Σ021 Σ0−1 11 Σ12 )

=



(σ 0 )2 1 − 23 (σ20 )2

   2 0 )2 0 σ0 (σ24 σ23 0 24 1− − σ34 − (σ20 )2 (σ20 )2

(C.4)

and 0 det(Σ022 − Σ021 Σ0−1 11 Σ12 )

0 2 (σ34 ) −2

>0⇔



(σ 0 )2 1 − 23 (σ20 )2

   2 0 )2 0 σ0 (σ24 σ23 0 24 > 0 (C.5) 1− − σ34 − (σ20 )2 (σ20 )2

0 )2 0 )2 0 σ0 (σ24 (σ23 σ23 24 0 σ +

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