Spatial dependence in patient migration flows

ABSTRACT This study analyses whether higher efficiency performance of Ecuadorian hospitals attracts larger inflows of interregional patients to a given hospital and assesses the existence of spatial dependence in the patient migration network. We develop an innovative two-stage approach. In the first stage, we use conditional order-m estimations to obtain robust hospital efficiency. In the second stage, we apply a spatial Durbin interaction model to estimate the effect of hospital efficiency on patient migration flows and disentangle the spillover effects. We identify the positive effect of specialized hospitals’ efficiency in attracting patients from other regions alongside spillover effects in the migration dyad.


INTRODUCTION
In healthcare system analysis, patient choice of hospitals and the resulting patient mobility is a topic that has occupied a vast body of the literature over the past two decades (Balia et al., 2014).Models allowing patient choice of hospitals have a wide range of useful applications, both for government decision-making as well as in hospital governance (Lowe & Sen, 1996).In this context, Balia et al. (2014) state that the importance of assessing patient mobility is twofold.Firstly, the geography of patient mobility offers indications as to the actual level of services provided.This can be particularly useful given that the preferences of individuals are not perfectly observable.For example, patient outflows may reveal the possible inefficiency or low quality of public healthcare supply in a given region. 1 Secondly, flow imbalances across regions may destabilize and put pressure on local healthcare budgets.This kind of information can be useful for central planners and regional authorities interested in correcting inefficiencies in the system as well as improving healthcare system performance.Understanding the mobility patterns of healthcare consumers may represent an important tool for central government and regional planners to identify clusters of hospitals and take advantage of spillover effects to better allocate resources and improve the efficiency of the system.
In essence, patients move between regions because they want to obtain the best hospital treatment that the system can provide, or at least access better services than those offered in their locality.They can be expected to move when possible inefficiencies translate into longer waiting lists, but also when perceptions of the quality of local healthcare services are negative (Aiura, 2013;Balia et al., 2014).These movements might become permanent over time if marked asymmetries in healthcare provision are evident between different regions in a country (Balia et al., 2018).
There is a strand running through the literature which argues that by eliminating barriers of access to healthcare, and thus giving patients the ability to choose between hospitals, this creates a financial incentive for healthcare providers to compete among themselves, leading to improvements in quality of care (Bloom et al., 2015;Gravelle et al., 2014;Propper, 2012).This theory might hold in a country where the healthcare system is homogeneous across regions.However, when regional disparities are significant and persistent over time, high-income regions tend to offer a better quality of care.This motivates patients to move from low-to high-income regions seeking better treatment.In turn, the dynamic of such flows, closely related to the spatial pattern, may give rise to network effects often detected in the data and translating into a structure correlation, known in the literature as spatial dependence (Anselin, 2010).
An interesting context of analysis is provided by Ecuador, whose marked regional disparities offer us a framework of study that can allow us to understand the interregional and intraregional dynamics of patient mobility that may be driven by performance gaps between heterogeneous hospitals.
Like other Latin American countries, Ecuador has suffered a continuous process of deterioration of its public healthcare system, which has been exacerbated by the neoliberal reforms of the 1990s and the 2000s, resulting in a widening of the existing territorial disparities in the country.These disparities are evident in a concentration of healthcare resources in a few public hospitals (the high performers), which at the same time were located in developed regions. 2With the approval of the new constitution in 2008, healthcare reforms were also introduced to promote free access to medical care and an extension of social security coverage.This gave patients the possibility of choosing the hospital where they wished to receive treatment. 3At the same time, this resulted in an increased demand for medical attention, thus encouraging a behaviour of mobilization to seek treatment in developed regions.
As barriers to access were removed, patients were expected to seek better treatment in areas where they perceived they would obtain the best possible treatment, leading to increased patient mobility.Mobility in turn triggered an increase in patient demand, which can result in two different outcomes.On the one hand, higher demand drives competition among hospitals in the region, resulting in an improvement in the quality of care or more efficient use of resources in order to cope with demand.On the other hand, when demand for hospital treatment increases, hospitals become crowded and additional resources are needed to reduce congestion, eventually resulting in inefficiencies like longer waiting times and ultimately in the underprovision of public services such as healthcare (Aiura, 2013).Moreover, if developed regions are the receivers of a larger share of patients, one can expect that other adjacent hospitals may also receive patients driven by the demand for neighbouring healthcare providers' services. 4 So far, the literature on patient mobility has focused on identifying and measuring the effects of such determinants on patient flows, either between regions or between healthcare institutions, but to date there has not been an empirical study assessing the dynamics of interregional patient mobility on hospitals within a given region.Understanding these dynamics can help regional planners and hospital managers to better understand patterns of demand and patient flows not simply as interregional, but also that they are intraregional in nature.High-performing hospitals can be better prepared to anticipate potential boosts in demand generated by new reforms that enable greater health insurance coverage among the population or allow for free access to medical services.Individual hospitals can thus account for these increases in demand and plan accordingly, in order to improve capacity, medical staff, or technological endowment.Low-performing hospitals can also benefit from this, and enhance their medical resources as well, to increase their performance and avoid possible outflows of patients.Piedra-Peña (2023) emphasizes the important influence that patient mobility can have on the performance of any given public hospital in Ecuador and that of surrounding hospitals, given the spillover effects in hospital efficiency. 5Here we seek to understand the patterns of these patient flows and determine the extent to which performance gaps are causing people to move from different regions to be treated at a high-performing hospital, and what the repercussions are for the surrounding hospitals.
This study therefore analyses whether increased hospital efficiency performance encourages larger inflows of interregional patients to a given hospital and whether these are accompanied by larger inflows of patients for neighbouring hospitals in the region.In this context, the reasoning behind patient movement is as follows: in order to obtain the best treatment possible, patients can observe different hospital features that indicate hospital quality, such as lower mortality rates, length of stay, and higher numbers of patients treated.However, these factors are optimized by the best use of hospital resources such as beds or physicians.This process of optimizing different hospital outputswhich can increase the perception of qualitywith a given set of inputs is what makes a hospital more efficient.Therefore, we can make use of an index that can measure the use of hospital resources to produce hospital outputs and reflect the complementarities of this process.Thus, capturing both the perception of a hospital's quality as well as its performance.At the same time, if interregional patients cannot be treated in a given high-performing hospital they might decide to obtain treatment from another high performer, located in the same region.
So far, the literature on healthcare economics has focused on the measurement of the effects of hospital competition, patterns of access to hospital services, and the determinants of patient migration flows by accounting only for the spatial distance between hospitals or regions, using gravity models (e.g., Congdon, 2001;Moscelli et al., 2016;Varkevisser et al., 2012).A significant proportion of the literature concludes that the efficient healthcare performance of hospitals and regions is a strong driver of patient mobility.However, to our understanding, there has been no attempt to consider the possible spillover effects that give rise to higher patient migration flows to neighbouring hospitals.In this respect, our contribution to the literature is to provide a robust measure of hospital efficiency, consistent with economic theory, that allows us to identify its effect in attracting patients.In addition, if spillover effects in the patient migration network are significant, this measure can serve as a reliable tool for decision-making to identify key hospitals that attract demand and foster competition.
To that end, we take an innovative two-stage approach.In the first stage, we make use of the conditional order-m efficiency measurement proposed by Cazals et al. (2002), Daraio and Simar (2005) and Daraio and Simar (2007) to obtain robust efficiency measures for Ecuadorian public hospitals in 2014.This method is based upon the economic concept of Pareto efficient allocation and takes into consideration the effect of other environmental variables (related to the region) on hospital performance.In this way, we estimate one index that allows us to measure how efficiently hospitals make use of their resources in order to provide medical attention.Furthermore, the use of the conditional order-m model allows us to incorporate multiple inputs and outputs into the measurement of efficiency and exploit the complementarities among these different hospital features in the provision of healthcare, which cannot be captured if these features are considered in isolation or used to calculate different ratios, which in turn could lead to mixed results and complicate potential decision-making (Piedra-Peña, 2023).In the second stage, we address patient mobility flows with spatial interaction models proposed by LeSage and Pace (2008) and LeSage and Pace (2009), which factor in traditional origin-destination (OD) models, but also incorporate spatial lags of the dependent variable in order to account for spatial dependence, represented by flows from neighbouring regions in these models and allowing for endogenous interactions (i.e., global spillovers).In addition, we consider exogenous interactions arising from contextual effects, taking into account spatial dependence in the explanatory variables, and representing characteristics of the neighbouring regions and hospitals (i.e., local spillovers) (LeSage & Fischer, 2016).In this framework, modelling spatial spillovers allows us to consider not just the intensity of the movements between ODcaptured in the classical gravity modelbut also approximate the contagion effect on neighbouring regions or hospitals.In other words, we are capturing not just the push and pull effects that regional and hospital features exert on patients but also how these features affect the flows from surrounding regions to neighbouring hospitals.Thus, by modelling spatial spillovers we make use of a proper framework of analysis that allows us to represent territorial heterogeneity in a context where a contagion effect is present and therefore affects the whole network of patient migration, rather than simply origins-destinations flows.For policy-making, modelling these flows may enable the taking of decisions targeted not just toward particular regions or hospitals, but through identifying clusters where tailored policies can take advantage of these spillovers and thus save scarce public investment and hospital resources in a developing country such as Ecuador.
In the applied literature, these models have been used in cases where origins and destinations coincide (LeSage & Thomas-Agnan, 2015).However, this does not conform to our case: the list of origins (regions/cantons) differs from the list of destinations (hospitals). 6This calls for a modification in the econometric estimation which has been recently addressed by Dargel and Thomas-Agnan (2022) although, to our understanding, is yet to be applied.The additional contribution of our study therefore lies in this attempt to apply it to the context of Ecuador.
The presence of endogenous interaction effects and, therefore, global spillovers mean that patient flows between an OD pair directly affect one another. 7For example, a change in patient inflows travelling along a given OD pair, generated by variations in efficiency, potentially impact patient movements (i) originating from a canton and going to alternative hospitals, (ii) originating from alternative cantons to a given hospital or (iii) originating from alternative cantons and going to alternative hospitals.In contrast, exogenous interaction effects, i.e., local spillovers, imply that changes in the characteristics of neighbouring cantons or regions affect variations in patient flows across OD dyads.Taking once again efficiency as an example, the existence of local spillovers suggests a competition effect among hospitals within the canton, as the increase in neighbouring hospital efficiency implies a higher inflow of patients into the region.
Our results show that efficiency is a strong determinant of interregional patient migration.However, this effect is significant only when we consider specialized hospitals (as opposed to general hospitals).We observe significant global spillover effects in the form of patients travelling to neighbouring hospitals within a region and coming from neighbouring regions to the canton of origin.These findings represent a useful tool for policy makers.Future healthcare reforms need to be well controlled and implemented since they must take into account territorial differences, not just in terms of healthcare resources but also of the level of specialization.In Ecuador, specialized hospitals are concentrated in a few developed areas, and their performance affects patient flows from other cantons.Because spillover effects are present, other hospitals within the region appear to benefit from this inflow.Higher competition among hospitals could lead to higher quality of treatment (Gravelle et al., 2014;Longo et al., 2017), but it may also prove detrimental if greater inflows give rise to congestion effects.Furthermore, future public investment in healthcare services could target clusters of hospitals in low-income regions which are likely to be the origin of patient migration flows toward high-performing hospitals.A sustainable strategy could be to support the construction of more specialized hospitals, or the opening of specialized wards in existing ones, and focus on incentives to drive local hospital competition so as to reduce gaps in the quality of healthcare with respect to high-performing hospitals.
This study is structured as follows.Section 2 reviews the literature on hospital patient migration.In Section 3, the theoretical model is described, as introduced by Brekke et al. (2016), which is followed throughout this study.Section 4 explains the methodology of the order-m efficiency measurement and the spatial interaction model.Section 5 describes our dataset and Sections 6 and 7 present the results and robustness analysis.Finally, the main conclusions are presented in Section 8.

LITERATURE REVIEW
The aim of our study is to identify the effect of hospital efficiency on interregional patient mobility.Moreover, we seek to disentangle the potential spillover effects found in these mobilization flows both between and within regions in order to define demand patterns in healthcare treatment that can potentially be used as a tool for decision-making.In doing so, we combine two different existing strands in the literature: healthcare efficiency measurement and patient mobilization.There is an extensive body of literature on healthcare efficiency measurement that focuses on obtaining a single value that measures the efficiency performance of an observed unit through parametric and non-parametric methods and combining multiple inputs and outputs.The idea of efficiency is linked to the concept of Pareto efficient allocation, where those efficient units are either minimizing inputs or maximizing outputs in the production of health (i.e., in providing medical attention).The main advantage of these approaches is that we can rely on a single estimated efficiency score, more consistent with economic theory, as it allocates technical or Pareto inefficiencies instead of measuring efficiency based on single averages (Cantor & Poh, 2018).A comprehensive review of this literature can be found in O 'Neill et al. (2008), Hollingsworth (2008) and Cantor and Poh (2018).
Furthermore, we rely on the hypothesis that the performance of a given set of hospitals is likely to be determinedto a certain extentby regional characteristics, and in particular by the level of development or income in the region (Brekke et al., 2016), due to the evident territorial disparities in Ecuador.In order to estimate efficiency scores that introduce environmental variables as a constraint of hospital performance, the applied literature indicates that they can be assessed in either one-step or two-step estimation models.The main setback of two-step approaches relies on a separability condition between the input-output space and the space of the contextual factors, assuming that these have no effect on the production process (Daraio & Simar, 2007).To avoid the separability assumption and provide meaningful results, we implement a non-parametric method known as the conditional order-m efficiency estimation (Cazals et al., 2002;Daraio & Simar, 2005).Recent applications of this technique include Halkos and Tzeremes (2011), who perform a conditional order-m efficiency analysis on Greek prefectures, and find a negative relationship between per capita gross value added (GVA) and efficiency, while population density has a positive effect on hospital performance.Other micro-level approaches, such as Mastromarco et al. (2019), analyse the cost efficiency of hospitals in the Czech Republic during the period 2006-2010.They implement an order-m efficiency estimation, controlling for non-profit status, teaching status, presence of a specialized centre (in the hospital) and occupancy rate, finding that non-profit hospitals, university hospitals and hospitals with specialized centres are generally less efficient.Another advantage of conditional order-m estimation is that we do not need to assume a production function in the estimation process.This is particularly important in our study, as the multidimensional nature of public hospitals and regional heterogeneity in the country posits a difficulty when defending the assumption of a single production function for all hospitals in the sample.
However, despite the clear advantages of these methods in providing a robust estimation of efficiency, there has been no attempt to combine them with econometric models to study patient mobility patterns.The empirical literature directly focused on patient mobility has been developed in the past two decades.Instead of focusing on specific determinants of patient flows, it concerns itself with modelling hospital choices and flows across different jurisdictions (Balia et al., 2014).Some micro-level studies single out potential determinants of mobility.Victoor et al. (2012) carried out a survey and found that some common determinants of patient mobility include patient characteristics (e.g., education, income and age) as well as provider characteristics.
They classify the former into structure indicators (which concern the organization of healthcare), process indicators (which relate to the care delivery process), and outcome indicators (which indicate the effect of the care delivered).In most of these studies, the performance of a hospital has been represented by basic productivity indexes and capacity indicators.
In our setting, we need to take into account regional macroeconomic variables since they impact patient decisions about seeking care across regional borders.Applied macro-level economic studies have mainly been based on gravity models, commonly used to model flows that take many forms, such as population migration, commodity flows and traffic flows (Thomas-Agnan & LeSage, 2014).These models embed movements of individuals between origin and destination regions.Levaggi and Zanola (2004) look for the determinants of net patient flows from certain regions of Italy to the rest of the country.They estimate gravity models for a sample of Italian regions from 1995-1997 and conclude that regions characterized by lower outflows are the ones that provide better or faster services.Cantarero (2006) develops the same analysis applied to patient flows across regions in Spain between 1996 and 1999 and identifies that patients from economically lagged regions tend to move more than those in regions that provide better health services.Fabbri and Robone (2010) explore the 'trade' phenomenon in hospital care, exploring the role of economies of scale and the impact of the north-south economic divide on patient mobility and Italian local health authorities, controlling for patient push and pull factors related to origin and destination.They find that richer local health authorities have a higher probability of attracting more patients, who present the most severe cases.
However, the use of traditional gravity models to explain spatial interaction can be limited.These models rely on a function of the distance of the OD to clear spatial correlation and cross-section independence.As LeSage andPace (2008, 2009) argue, the use of distance functions to effectively capture the spatial dependence of observations may be erroneous, as well as the notion that flows are independent, since OD flows are fundamentally spatial in nature.In our framework of analysis, we expect to find a behaviour pattern where high-income regions are the main receivers of patients, following a spatial pattern, that, if not controlled for in the econometric estimation, could lead to biased conclusions.
So far, no studies have tried to account for spatial dependence in patient mobility.Moreover, even though a significant proportion of the literature implicitly concludes that healthcare performance is a strong driver of patient flows, there has been no attempt to disentangle its sole effect.There is however one study by De Mello-Sampayo (2016) that uses length of stay as a proxy of hospital performance and quality to analyse hospital choice in mental health patients in Texas, controlling for geographical patterns.In doing so, they build two indices of hospital competition and accessibility and conclude that patient flows depend not just on push, pull and spatial factors, but crucially on the geographical pattern.However, they rely on different estimations including differing combinations of covariates controlling either for competition or accessibility, and while the direction of their effects is robust in all specifications, the size is not always comparable, which complicates any potential decision making.The closest paper to our approximation is Balia et al. (2018), who account for local spillover effects by incorporating the spatial lags of the exogenous variables into the gravity model.They use a spatial panel data framework of Italian hospital discharges between 2001 and 2010 to assess the effect of the main determinants of interregional patient flows, differentiating between the impacts of regional health policies and other exogenous factors.Their results show that supply factors in neighbouring regions, alongside specialization and performance, largely affect mobility by generating local externalities that explain OD patient flows, thus bringing some novel insights into the inherent spatial-dependent nature of hospital performance and its effect on patient mobility.
Our empirical estimation therefore goes beyond the incorporation of local spillover effects, as per Balia et al. (2018), and includes potential global spillover effects likely to be found in OD flows, as stated by LeSage andPace (2008, 2009).In doing so, we use the extended gravity models developed by LeSage andPace (2008, 2009) to allow for spatial dependence in the sample, represented by the flows from cantons (i.e., regions) to public hospitals in these models.In addition, we consider exogenous interactions of the explanatory variables (LeSage & Fischer, 2016) to account for the contextual effect of the neighbouring regions and hospitals in the OD dyad, as per Balia et al. (2018).The introduction of endogenous and exogenous interactions into the econometric model allows us to take into consideration the spatial structure present in OD flow data that is not completely captured by the sole inclusion of the distance between origin and destination.If spillover effects are found to be statistically significant, then policy implications may be suitably directed to identify key players within the flow network exerting an indirect effect over other hospitals.Policy decisions can target those key players to improve the healthcare performance of the region.

THEORETICAL FRAMEWORK
In our framework of study, the high-performing hospitals are mainly located in developed regions (see Section 5) where historically healthcare resources in the country have been concentrated.These asymmetries in hospital performance are rooted in regional healthcare performance gaps that may incentivize those patients residing in less developed regions (i.e., cantons) to seek treatment in high-performing hospitals.The basis of our theoretical framework draws upon Brekke et al. (2014Brekke et al. ( , 2016)), who studied asymmetrical regions, where regions differ in their ability to provide healthcare services: the higher the performance gap between providers, the higher the number of patients who will seek medical care in high-income regions.Brekke et al. (2014) argue that patient mobility can play a significant role in efforts to improve welfare provision.However, such welfare improvements are often accompanied by further asymmetries.If competition encourages improved performance, then patients living in regions with high-performing hospitals are better off than under a healthcare system with restricted mobility.Conversely, in areas of lowperforming hospitals, only patients who move to high-performing areas benefit from the quality improvement in healthcare.In addition, Brekke et al. (2016) consider a framework with heterogeneous income levels across and within regions.They find that reducing barriers to free patient mobility represents an incentive to reduce quality in low-to middle-income regions, while increasing income disparities between regions as well as the gap in interregional quality.
We draw upon the cross-border patient mobility theory of Brekke et al. (2016).This theoretical model relies on the idea that, in equilibrium, regions with higher income offer better quality, which creates an incentive for patient mobility from lower to higher income regions.This concept can be applied to our setting, as the best-performing hospitals are mainly located in high-income regions. 8A detailed description of Brekke et al. (2016) model can be found in Appendix 2 in the online supplemental data.

METHODOLOGY
The method used in this study is developed in two stages.Firstly, we need to obtain the efficiency measures for each hospital, subject to the environmental variables they face and which can constrain their performance.In the second stage, we develop a spatial interaction model based on the conventional gravity specification to estimate the impact that the efficiency value has on migration flows, allowing for potential spillover effects.

Order-m efficiency estimation
The first stage of our strategy uses non-parametric order-m efficiency estimation, introduced by Cazals et al. (2002), Daraio and Simar (2005) and Daraio and Simar (2007).In comparison with other classical non-parametric approaches (such as data envelopment analysis (DEA)), order-m methods have two main advantages: (1) the estimations are robust even in small samples, and (2) they mitigate the impact of extreme values and outliers.The idea of order-m estimation is as follows: instead of considering the full sample at the moment of estimating the efficiency scores (like DEA), the order-m frontier considers as a benchmark the best practice expectations among m peers randomly drawn from the population of units. 9The procedure is repeated B times, resulting in multiple efficiency measures ( û1 m , . . ., ûB m ), where the final order-m efficiency value is the sample mean ( ûm ).In this way, the efficiency of a decision making unit (DMU) 10 can be compared with m potential DMUs that have a production larger or equal to y.In this study, we take an output-oriented approach, as we expect that, to a certain extent, patients can form their perceptions of hospital performance based on the outputs used. 11Our final efficiency index will take values ûm = 1 which correspond to technically efficient hospitals, while ûm . 1 are technically inefficient.Finally, we can also obtain values lower than 1, corresponding to super-efficient hospitals.

Spatial Durbin interaction model
In the second stage of our strategy we make use of spatial interaction models, which rely on gravity models to explain OD migration flows. 12However, a potential drawback for gravity models is that they rely purely on a function of OD distance to account for spatial correlation and ensure cross-section independence (Balia et al., 2018).These assumptions have been challenged by many authors.Porojan (2001) and Lee and Pace (2005) find evidence of spatial dependence in the residuals of international trade and retail sales flows, while LeSage and Pace (2008,2009) argue that the assumption of independence among observations may be difficult to defend, as OD flows are fundamentally spatial in nature.The explicit consideration of flow data correlation due to the spatial configuration of the units involved has attracted much attention in the literature, often referred to as network autocorrelation (Patuelli & Arbia, 2016).To embed spatial dependence in a spatial interaction setting, LeSage and Pace ( 2008) argue that it is necessary to consider spatial spillovers in three dimensions: origin-based, destination-based and origindestination based.As per LeSage and Pace (2009), origin-based spatial dependence is where forces leading to flows from any origin to a particular destination may create similar flows from neighbouring origins.Destination-based spatial dependence is the idea that forces leading to flows from the origin to a destination may generate similar flows to nearby destinations.Origin-destination-based spatial dependence describes those forces that create flows from neighbouring regions of the origin, to neighbouring regions of the destination.Using this definition of spatial dependence means that we need to model spatial dependence for flows of patients as a spatial autoregressive specification, accommodating endogenous interactions.This definition will allow us to define spatial spillover effects to hospitals neighbouring the destination hospital in the flow of patients.In addition, we can accommodate the model for exogenous interactions in a spatial Durbin model (SDM) representing a situation where local spillovers arise due to changes in the characteristics of neighbouring hospitals and the environmental features of neighbouring regions (i.e., cantons).Exogenous interactions can be modelled by including the spatially lagged covariates in the econometric specification (along with the spatial lag of the endogenous variable).If statistically significant, the omission of these interactions can lead to problems of omitted variable bias (LeSage & Fischer, 2016).We control for this issue and begin by defining the model with no spatial interactions (based on the conventional gravity model) and adjust it for exogenous interaction specifications as per LeSage and Fischer (2016).Then, we move to its SDM extension as described in LeSage and Pace (2008) and Dargel and Thomas-Agnan (2022). 13We define our final model as follows: where y is the N = n o × n d vector of logged patient migration flows, obtained by stacking the columns of the matrix Y of patient migration flows, whose columns indicate their origins (i.e., cantons), with the rows representing destinations (i.e., hospitals).In this, n o is the number of geographical observations at the origin and n d the number of geographical observations at the destination.The vector of distances g is composed by the Euclidean logged distances between origins and destinations.We define a contiguity matrix W d where hospitals are neighbours if they are located in the same canton.Therefore, W o is defined as neighbour of those cantons that share a border line.W w is defined as the product of the two weight matrices (W o .W d ).The spatial lag r d W d y reflects flows from neighbours to each destination observation in the vector of origin-destination flow to form a linear combination of flows from neighbouring destinations, while r d captures the strength of destination-based dependence.Similarly, r o W o y reflects a linear combination of flows from regions neighbouring the origin, and r o reflects the strength of originbased dependence.Consequently, r w W w y forms a linear combination of flows from neighbours to the origin and flows from neighbours to the destination, with the parameter r w representing the strength of this dependence.
The vector e d contains the robust logged efficiency scores specific to every hospital.The matrix X o accounts for economic and demographic regional characteristics that represent regional income levels and health conditions and impact patient decision-making and choice with regard to seeking treatment in other (developed) regions.We use cantonal variables such as logged gross value added per capita (GVApc), logged population density, logged cantonal mortality, logged unsatisfied basic needs index (NBI), 14 and logged insured population rate. 15β o , b d are the associated k × 1 parameter vectors.The scalar parameter g is the effect of the (logged) distance g, and a is the constant with l N vector of ones.
The spatial lags of the exogenous variables W o X o and W d X d help explain variations in flows across dyads coming from changes in the characteristics of regions neighbouring the origin and hospitals neighbouring the destination, respectively.Ø o , Ø d are the parameters associated to W o X o and W d e d .We control for the spatial lags of distance g, with W dg and W o g describing variations in flows arising from changes in the distance of neighbouring hospitals within the same canton and from neighbouring cantons, respectively.This aligns with the idea that patients will select a hospital for treatment on the basis of not only their proximity to a given hospital, but also to that of neighbouring hospitals.Finally, g d and g o are the parameters corresponding to W d g and W o g; and 1 is the vector of disturbances.
A problem that may arise in the application of the model (especially for the regions demonstrating a large inflow of patients) is the presence of large flows of patients in the matrix of OD flows, relative to smaller (or even zero) flows.This would produce non-normality in flows and jeopardize the estimations (LeSage & Pace, 2008, 2009;Thomas-Agnan & LeSage, 2014).In our setting, this represents an intraregional flow of patients (e.g., residents of developed regions obtaining treatment in their local area).To deal with this issue, LeSage and Fischer (2010) propose a modification of the independent variables, by replacing the values of the independent variables representing intraregional flows with zero.Intraregional variations are thus captured in a new set of explanatory variables X i , W i X i , with non-zero observation for the intraregional observations alongside the addition of a new intercept term, a i .
The model can be estimated using maximum likelihood (see LeSage & Pace, 2008).LeSage and Pace (2009) also show how to produce Bayesian Markov chain Monte Carlo (MCMC) estimates for the model. 16In this study, we follow the Bayesian approach using the computational methods proposed in Laurent et al. (2019).
Note that we cannot interpret b d (nor any other estimated parameter associated with origin-destination characteristics) as the partial derivative on flows originating from changes in the destination-efficiency.As pointed out by LeSage and Pace (2009), in the spatial econometric specification of the interaction model, changes in the kth characteristic of an observation i will subsequently produce changes in flows into the ith observation from other observations, as well as flows out of the observation i to other observations, unlike conventional regression models where this leads to changes only in observation i of the dependent variable, y i .
LeSage and Thomas-Agnan (2015) propose scalar summary measures of the impacts arising from changes in characteristics of the observations that involve averaging the cumulative flow impact associated with changes in all observations, resulting in what are referred to as origin effects, destination effects and network effects.Origin and destination effects express the mean impact on flows arising from changes in origin and destination characteristics, respectively.In turn, network effects characterize the mean impact of a change in characteristics of the origin i on all the flows originating from other origins, different from i to a destination j.

DATA AND VARIABLES
To estimate Equation (1), we collected data pertaining to Ecuadorian public hospitals for the year 2014.We are interested only in public hospitals for two main reasons.Firstly, the healthcare reforms implemented alongside the new constitution in 2008 with the dual objectives of extending provision of free medical services and social security coverage only affected public hospitals.Hence, populations looking for better medical attention are only able to obtain it from public healthcare providers.Secondly, to have access to private healthcare, patients would require a private insurance or pay out of their own pocket (which would entail high levels of expenditure if specialized care is needed).Moreover, just 6% of the population were covered by private insurance in 2014 (Survey of Life Conditions).
Hospital information was obtained from the Annual Survey of Hospital Beds and Discharges and the Survey of Health Activities and Resources provided by the National Institute of Statistics and Census (INEC, Spanish acronym).We excluded psychiatric, dermatology and geriatric hospitals, and removed from the sample those hospitals that were likely to present irregularities in the data. 17As described above, the migration flow matrix considers the rows to represent the hospital destination, while the columns are the cantons (i.e., regions) of origin.We retrieved a sample of 176 destination hospitals and 106 cantons of origin.By vectorizing the flow matrix, we obtain a vector of 18,656 observations.
The cantonal economic and demographic variables were retrieved from the Ecuadorian Central Bank (BCE, Spanish acronym) and INEC's public statistics, while the poverty and population insurance data was collected from the 2010 national census.The description of all the variables is presented in Appendix 6 in the online supplemental data.

Variables for the conditional order-m efficiency measurement
For the selection of input and output variables in the first stage of our strategy, we followed the existing literature on efficiency measurement.A complete review of the literature is offered in Hollingsworth (2008); O'Neill et al. ( 2008) and Cantor and Poh (2018).
Regarding the input variables, we use the number of beds, medical equipment and infrastructure, widely used as a proxy for hospital size and capital investment (O'Neill et al., 2008).To represent labour costs, clinical staff is also usually included (Hollingsworth, 2003(Hollingsworth, , 2008)).To this end, we include both the number of physicians as well as other clinical staff members at the hospital.
As for outputs, we use hospital patient discharge records to measure the final provision of healthcare.To control for the heterogeneity of cases attended to, we build a case-mix.This index is used in the healthcare efficiency measurement literature to control for the severity of cases treated, as not all the patients can be treated using the same amount of resources, nor can all hospitals possess the means to treat the most severe cases (Cantor & Poh, 2018).We use the case-mix index proposed by Herr (2008), which relies on the assumption of a positive correlation between length of stay and severity of illness.The index is built according to the three-digit International Statistical Classification of Diseases and Related Health Problems (ICD-10).
Furthermore, the 2014 Survey of Health Activities and Resources provides information on the total number of morbidity and emergency consultations, as well as the total number emergencies treated, both commonly used in the literature to measure hospital activity (Cantor & Poh, 2018).
In addition, we aimed to take into account a hospital-related quality output, referring to the patient survival rate 48 h after admission to hospital.This is because the mortality rate after 48 h indicates the resolutive capacity of hospital employees.Therefore, it has a higher correlation with the quality of treatment provided.Hospital mortality rates have been frequently employed to represent the quality of hospital treatment (Hollingsworth, 2008).However, in healthcare efficiency measurement we need to measure the outputs as patient health gains, which is why the survival rate (1-mortality rate) is usually employed. 18 In this respect, other hospital indicators such as readmission rates, the level of specialization (Gravelle et al., 2014;Longo et al., 2017) or nosocomial infections (Prior, 2006) have been regularly employed to represent hospital quality.However, we do not account for this information in our dataset, which is one of the limitations that we encountered in our study.
Finally, we consider three environmental variables that can potentially affect hospital performance: cantonal GVApc, cantonal population density and hospital occupancy rate.The first two explain territorial inequalities in the country, which have a major influence on patterns of regional development (Mendieta Muñoz & Pontarollo, 2016).The developed regions contain a high concentration of hospitals and health resources, which strongly influences healthcare performance.Piedra-Peña and Prior (2023) find empirical evidence that these developed regions do not just feature concentrations of better-quality hospitals, but that these hospitals are also the best performers in terms of efficiency.The empirical evidence of the effect of GVApc and population density on healthcare efficiency is supported by Halkos and Tzeremes (2011).
The third environmental variable, hospital occupancy rate, is commonly used to represent the utilization of potential capacity in a hospital and determine whether it adjusts clinical staff levels in response to increases in numbers of patients treated in the short run (Herwartz & Strumann, 2012, 2014).The idea behind this is that hospitals with low occupancy rate may be overstaffed, and thus inefficient.Occupancy rate has been used as an environmental variable for conditional order-m approaches in recent work by Mastromarco et al. (2019).Furthermore, Piedra-Peña (2023) provides empirical evidence of both its positive direct and spillover effects on hospital efficiency.
Table 1 presents the descriptive statistics of variables included in the conditional order-m efficiency estimation.Overall, we can distinguish a significant disparity in hospital inputs and outputs (as well as in cantonal variables), observed in the difference between the minimum and maximum values describing the marked discrepancies across hospitals and cantons.In fact, Piedra-Peña (2023) emphasizes that those hospitals endowed with a high level of resources and numbers of patients treated tend to be concentrated in densely populated regions and with high production levels (measured by the GVA).This initial evidence supports our hypothesis that patient movement is likely to be directed to those developed regions, where more healthcare resources are concentrated.

Variables for the spatial Durbin interaction model
Firstly, at the hospital level, we use the efficiency scores obtained in the first stage as a variable of hospital performance.This variable represents the pull effect of a hospital in attracting patients (e d in Equation ( 1)).A negative sign of the efficiency variable destination effect indicates good performance, attracting patients from other cantons. 19The rationale for this is twofold.On the one hand, patients identifyto a certain extentthe best performing hospitals, opting to travel another other canton (potentially the developed ones) to obtain treatment at what they perceive to be the best facility.Therefore, the efficiency performance of a hospital also explains the patient perceptions of quality.On the other hand, this inflow of patients can also be explained by referrals from low-tech hospitals that may not have the requisite resources to treat a complex pathology.Unfortunately, we do not have access to information regarding hospital referrals in our dataset to test this hypothesis.However, in both hypotheses, the significant effect of efficiency performance helps to explain interregional patient mobility and quality perceptions either by the patient or the hospital that refers the patient (or both).
We represent the cantonal level variables (X o in (1)) that will impact on patient decisions to seek medical treatment with five variables.Firstly, we use GVApc and population density to indicate the level of development in the region.As in the first stage, these variables can explain the regional heterogeneity that characterizes the country.As a result, it is very likely that the most indemand hospitals where both population and economic activity are more concentrated are located in developed cantons, which, over time, can foster quality differentials (Balia et al., 2020).This observation has been empirically demonstrated in Ecuador (Piedra-Peña, 2023;Piedra-Peña & Prior, 2023).The use of these variables to explain patient mobility has been extensively applied in the literature (see for example Cantarero, 2006;Fabbri & Robone, 2010).We use cantonal mortality (per 1000 individuals) and the insured population rate to represent healthcare conditions in the region, and control for accessibility to medical treatment.The intuition is that higher mortality rates are associated with poorer healthcare conditions in the canton.
Finally, we control for the poverty level in the canton by introducing the unsatisfied basic needs index (NBI).The index was developed by The Economic Commission for Latin America and the Caribbean (ECLAC, or CEPAL with the Spanish acronym), and has been widely applied in Latin American countries since the 1980s as a multidimensional measurement of poverty (CEPAL, 2007).Considering that poverty is a complex and multidimensional phenomenon, the NBI evaluates different dimensions of deprivation of goods and services required to the satisfaction of basic needs.In Ecuador, these dimensions comprise economic capacity, basic education access, housing access, basic services access, and overcrowding.As stressed in the theoretical model, richer patients are more prone to opt for cross-border healthcare.We therefore attempt to represent this dimension of regional patient heterogeneity with the poverty index.Table 2 presents the descriptive statistics of the variables used in the SDM model.In addition, Figure 1 shows the distribution of hospitals by efficiency performance (e d in Equation ( 1)) in the top panel (a), and the migration flow dynamic of the sample (y in Equation ( 1)) in the bottom panel (b). Figure 1a shows the most efficient hospitals (i.e., hospitals with an efficiency value lower than 1) to be mainly concentrated in two of the most developed regions of Ecuador, where most of the country's healthcare resources are located. 20 Figure 1b shows patient flows from origin to destination, organized by intervals.We observe that there is a clear dynamic of patients travelling to the regions where the best performing hospitals are concentrated.We can appreciate that most of the patient inflow comes from neighbouring cantons, representing an initial indicator of potential spatial autocorrelation in the migration flow.Hence, we use spatial interaction models that allow us to disentangle the spillover effects of this migration dyad.

RESULTS AND DISCUSSION
The estimation results of our SDM model laid out in Equation ( 1) are presented in Table 3.The Bayesian MCMC estimates are based on 1000 draws.Lower and upper 0.05 and 0.95 credible intervals are reported, as well as the t-statistic. 21 The estimates show not only a high level of destination-based spatial dependence, but also origin-based and origin-destination-based spatial dependence.The coefficients r d and r o are 0.31 and 0.53, respectively.The estimated parameter r w is −0.11 and statistically different from zero.The 95% intervals suggest a small standard deviation and therefore, a high level of precision in the estimation.
These results provide evidence of the existence of spillover effects as a result of patient migration flows.Destination-based spatial dependence posits that flows coming from a given canton of origin to a destination hospital creates similar flows to neighbouring hospitals (located in the same destination canton).In addition, origin-based spatial dependence shows that flows from any canton of origin to a destination hospital create similar flows from neighbouring areas.Finally, origin-destination spatial dependence demonstrates that larger outflows from cantons neighbouring the origin indeed generate larger inflows to hospitals neighbouring the destination. 22These findings indicate the existence of spillovers occurring not just between cantons, but also within cantons.
As per Thomas-Agnan and LeSage (2014) and LeSage and Thomas-Agnan (2015), the coefficients and t-statistics reported in Table 3 should not be interpreted as reflecting the partial derivative effects of changes in origin and destination characteristics.In turn, we need to calculate origin,  destination and network summary measures to draw valid inferences on how changes in origin and destination characteristics impact the decision-making process driving patient migration flows.Table 4 summarizes the scalar summary effects in Equation (1).In terms of hospital efficiency, the estimates show a significant expected negative effect.The increase in efficiency of an observed hospital leads to a higher inflow of patients.Specifically, a 1% increase in the efficiency of an average hospital would lead to a 0.3% increase in patient inflows. 23As previously mentioned, these results support the hypothesis that patients perceive those hospitals that indicate a higher performance as more qualified and competent.Higher efficiency performance appears to function as a pull factor attracting patients from neighbouring regions.This effect can also occur as a result of patient referrals from other (low-performing) hospitals, which are not endowed with the necessary resources to treat complex pathologies.The information available in our dataset does not allow us to disentangle the size of these effects.We leave this question to be explored in future research.Interestingly, hospital efficiency also displays a significant negative network effect.This means that a 1% increase in the efficiency of a given hospital leads to an increase in patient movements to neighbouring hospitalsdifferent to their initially preferred hospital destinationof 0.15%.These findings are in line with Piedra-Peña (2023), suggesting a competitive effect where higher efficiency in neighbouring hospitals increases patient inflows.
Changes in the characteristics of the canton of origin provide additional information on patient travel decisions.For example, the positive and significant impact of GVApc could provide a measure (ceteris paribus) of the wealth effect of the origin.If the GVApc of the canton of origin increased, patients would have more disposable income available to devote to travel costs incurred when seeking medical treatment in other regions, thus creating a push effect in that region.The positive and significant network effects of GVApc point to an increase in outflows from cantons neighbouring the origin when their level of wealth increases.This supports our assumption that regional income levels are a key determinant of cross-border patient migration, as stated in Section 3.
Furthermore, densely populated cantons with high mortality rates are expected to push patients away.However, it is interesting to observe a negative network effect for both these variables.An explanation of the latter could be that high density and mortality in a neighbouring canton disincentivizes patients from seeking treatment in other regions different to their place of origin.
Before drawing any conclusions, we need to test the robustness of our results.We thus provide a robustness analysis in Section 7.

ROBUSTNESS ANALYSIS
In order to check the robustness of our results, we carried out a range of tests.A detailed explanation of all the robustness checks is presented in Appendix 8 in the online supplemental data.Firstly, we test the sensitivity of our efficiency estimator against different values of m peers.We do not find significant differences within the range of the m value selected in this study.We also checked for the validity of our environmental variables, finding evidence that hospital performance is indeed affected by regional income levels, and this effect is captured with the order-m conditional model.
In the second stage, we checked for potential endogeneity in our efficiency variable.We do not find evidence of a correlation between this variable and the error term of our econometric specification.Furthermore, we do not find any bias in our estimations when we consider information from the previous year, nor when we exclude emergency consultations from the efficiency estimation (given that these are not always under the control of the hospital).
Moreover, we checked for the sensitivity of the results to alternative specifications of the spatial weight matrix.We find comparable results for origin and destination effects.Finally, we check the robustness of the results by considering the spectrum of treated diseases.There is the possibility that the pull effect could mainly be driven by the presence of specialized hospitals as opposed to other general hospitals that provide a different range of treatments.Thus, we split the sample into two different subgroups by distinguishing between general and specialized hospitals (the latter subgroup includes chronic and acute hospitals). 24 Table 5 presents the scalar summary effects for efficiency and the parameters ρ - d ρo .and r w for each hospital type.It is not surprising to observe that the destination effect for general hospitals disappears, suggesting that the pull effect of hospital efficiency performance is mainly captured by specialized hospitals, as the magnitude of the estimation is greater.As general hospitals spread across the country, what appears to be driving people to travel to high-income regions is the performance of specialized medical institutions, which are more concentrated in more affluent cantons (see Figure 5 of Appendix 9 in the online supplemental data).However, the high performance of an average general hospital is not sufficient to attract interregional patients, as they are likely to seek medical attention at their local hospital to treat a common disease.Instead, in the case of specific or severe illnesses, patients will select a particular hospital on the basis of the quality of the treatment they perceive they will receive there, which is captured by our efficiency variable.Nevertheless, spillover effects remain statistically robust and comparable in both cases, meaning that both arrangements are valid and thus guarantee patient mobility across the territory.One explanation endorsing these results (particularly for general hospitals) is that even though the increase in efficiency of a given hospital is not enough to attract intraregional patients, those hospitals take advantage of patient inflows, initially attracted by other hospitals (most likely neighbouring specialized hospitals). 25 Our results open up an important discussion in terms of implications for policy.Hospital efficiency performance appears to play a significant role in shaping perceptions of quality among public hospital patients, which cannot be overlooked.In this respect, policy makers need to take into consideration that the unexpected effects of implementing healthcare reform could include a range of consequences beyond the measures being addressed and directly or indirectly affecting those healthcare institutions initially targeted.For example, new reforms that remove barriers to access to more specialized and sophisticated treatment (only available at specialized hospitals) need to be well planned and allocated.If increases in demand driven by these reforms are not carefully monitored, this could lead to congestion effects that may impact the performance of specialized hospitals.Due to spillover effects, neighbouring hospitals (including general hospitals) could experience detrimental consequences, 26 leading to a deterioration in regional healthcare performance.
So far, Ecuadorian healthcare reforms have been accompanied by an increase in hospital efficiency, with hospitals allocating spare resources to deal with the higher inflow of patients.However, these reforms have been mainly focused on offering general treatment at public hospitals that are abundant and spread quite homogeneously across the country.There is a lower supply of specialized hospitals which are much more territorially concentrated.These findings highlight the importance of implementing tailored regional healthcare policies.
As Brekke et al. (2014) suggest, high-income regions could also be benefiting from welfare improvements, as we identified a competition effect in efficiency among hospitals within the same regions, leading to higher regional performance and quality.However, welfare effects could also generate asymmetries, as low-income regions are not endowed with high-performing specialized hospitals, and only patients moving to other regions benefit from these welfare improvements.Future public investment could be focused on increasing specialized services for hospital clusters in low-income areas.An increased supply of specialized hospitals could attract patients and drive competition among hospitals to provide welfare improvements and reduce quality disparities between regions.

CONCLUSIONS
This study aims to analyse whether the higher efficiency performance of Ecuadorian public hospitals results in a greater inflow of interregional patients to a destination hospital, and whether this in turn leads to a higher inflow of patients to neighbouring hospitals within the same region.To determine the effect of efficiency on the patient migration network, we follow an innovative two-stage strategy where the first step is to estimate robust conditional order-m efficiency values, based on the economic concept of Pareto efficient allocation, with the second step making use of a spatial Durbin interaction model to estimate the effect of hospital efficiency on patient migration flows and separating spillover effects in the form of larger inflows of patients to neighbouring hospitals.We contribute to the applied empirical literature by estimating a model that considers different origins and destinations in the OD dyad, that, to the best of our knowledge, has not yet been applied.
We refer to a structure in which regional disparities are modelled by means of healthcare asymmetries over time, producing a healthcare performance gap across regions and motivating a patient mobilization pattern, with the bulk of the influx of patients concentrated in developed regions.Our results support the hypothesis that hospital efficiency performance is a strong pull factor affecting this inflow, and the direction of this effect is robust according to different specifications and estimation methods.However, when we split the sample separating general and specialized hospitals, this effect disappears in the former, but becomes more pronounced in the latter.In addition, we identify spillover effects in mobilization flows, not just in the form of patients arriving at neighbouring destination hospitals from a canton of origin, but from patients arriving at a given hospital from cantons close to that origin, as well as arriving at adjacent hospitals.
This evidence has two implications.Firstly, the efficiency effect suggests that, to some extent, patients perceive hospital performance as a proxy for hospital quality that in turn encourages cross-border migration in search of better medical treatment than that which is available in their local area.However, this decision is based on the availability of specialized hospitals in the destination region, which are mostly concentrated in highly developed areas.The possibility also exists that other hospitals refer patients with complex diseases, as they do not possess the resources to treat them.Secondly, spillover effects evident in the data suggest that other hospitals neighbouring the specialized institutions also capture some of those patient inflows.According to Brekke et al. (2014), if there were competition among hospitals (which we indeed identify with the statistical significance of the network effects), this could have a beneficial effect on the welfare of the population, as more competition encourages higher quality of care.However, hospitals in less-developed regions might not benefit from welfare increases, as there is no incentive to provide better medical attention and therefore only patients travelling beyond regional borders may benefit from it.
Our results deliver useful suggestions for policy makers.On the one hand, new reforms need to be well-planned, not just in terms of territorial discrepancies but also in terms of hospital specialization.For example, decreasing limitations to specialized care could trigger an increase in healthcare demand, that, if not controlled, could entail negative consequences such as congestion effects.Negative shocks to specialized hospitals have a negative impact on their performance as well as on demand for hospitals in the surrounding area and, as a consequence, this adversely affects the efficiency of hospitals in the whole region and, consequently, the welfare of the population.Public authorities could identify the key players in the healthcare network to target reforms that could encourage better performance within the public healthcare system of the region due to spillover effects.
Public healthcare policy can devote a larger share of resources to targeting investment in lessdeveloped regions.The significant level of origin-based spatial dependence suggests the existence of clusters of less-developed cantons recording an outflow of patients.If there were not enough demand for local hospitals to compete, there would be no incentive to increase the quality of care.Therefore, public investment could be focused on the creation of specialized hospitalsor specialized wards in existing hospitalsin these regions to attract more demand.Once the inflow of patients is established, new spillover effects could occur, benefiting adjacent hospitals and bringing improvements both in regional healthcare performance and welfare, so as to benefit the low-income patients of that place, who cannot afford to receive treatment in other cantons.
Finally, future research possibilities can be derived from this study.As indicated, the effect of efficiency performance on migration flows could be driven by the perception of patients when selecting a given hospital (i.e., where they perceive they could receive better medical treatment) or by other hospitals referring highly complex cases to those best-performers (or indeed both).Unfortunately, our dataset does not account for information on patient referrals to disentangle the size of these effects, but it opens up interesting methodological research strategies to be investigated in future studies.attention derived from given outputs (e.g., patients treated, treatments carried out).In this sense, a fully efficient hospital can maximize its outputs with a given amount of inputs.This is commonly known in the healthcare efficiency measurement literature as technical efficiency (Hollingsworth, 2008). 2 Refer to Appendix 1 in the online supplemental data for a description of the Ecuadorian healthcare institutional framework 3 The 2008 constitution (which declared that health is a basic right and committed the state to ensuring the full exercise of this right alongside more extensive social insurance coverage) initiated reforms intended to enable increased access to medical treatment, such as the free medical services provided by the Public Ministry of Health (MSP) and saw the implementation of laws penalizing employers who do not affiliate their employees to the social security system (Orellana et al., 2017). 4For example, if a given hospital has a long waiting list, patients may try to receive attention at alternative hospitals in the region. 5Piedra-Peña (2023) provides evidence of the existence of positive spatial dependence in public hospital efficiency due to the existence of global and local spillover effects.In other words, an increase in the efficiency of neighbouring hospitals also has a positive impact on the efficiency of an observed hospital. 6In Ecuador, cantons are second-level administrative divisions.The Republic of Ecuador is divided into 24 provinces, which in turn are divided into 221 cantons.The cantons are further subdivided into parishes. 7Hereinafter, we will refer to cantons (or regions) as the origin observations of our OD dyad.Conversely, hospitals will be referred to as the destinations of the OD dyad. 8Brekke et al. ( 2014) develop a Hotelling model with two regions that differ in healthcare technology, where regions with more efficient technology supply higher-quality healthcare, attracting patients from neighbouring regions with less efficient technology.However, the restriction of incorporating two regions prevents the consideration of cases where a region can be both importing and exporting patients, as opposed to Brekke et al. (2016) (who incorporate a three-region specification).In addition, the framework used in Brekke et al. (2016) allows for extra expenses when patients demand care outside their region, as well as for heterogeneity in income within regions (with wealthier patients more likely to move). 9We fix the value of m = 90, as per the approach of Daraio and Simar (2005) for which the percentage of super-efficient DMUs decreases smoothly with an increase of m. 10 We can apply the term DMU to any unit of analysis e.g., individuals, departments, firms, municipalities, or in the case of this study, hospitals. 11The reader can refer to Appendix 3 in the online supplemental data for a more in depth explanation of the order-m methodology followed in our approach. 12Refer to Appendix 4 in the online supplemental data for a detailed description of the LeSage and Pace (2008,2009) spatial interaction model. 13We take this step to identify the sources of spatial autocorrelation and in order to avoid model misspecification and omitted variable bias.Following this sequence, we can determine the significant effect of exogenous interactions by means of Spatial Lagged X (SLX), and, then, those of the endogenous interactions with the SDM model.In such a way, we can select the appropriate framework of analysis that provides the best fit to our data. 14The NBI is a multidimensional poverty index, commonly used in Latin American countries. 15In order to avoid taking the log of zero, we have added the unity to the dependent and independent variables as per LeSage and Thomas-Agnan (2015). 16Refer to Appendix 5 in the online supplemental data for an explanation of MCMC estimation in LeSage and Pace (2009).

Figure 1 .
Figure 1.Hospital efficiency and patient migration flows.

Table 1 .
Summary statistics of the conditional order-m variables.
Note: GVA expressed in US dollars.Source: The authors, based on information from INEC and BCE.

Table 2 .
Spatial interaction model variables: summary of statistics.
Note: GVA expressed in US dollars.Source: The authors, based on information from INEC and BCE.
Note: Dependent variable is the vector of logged migration flows.Bayesian MCMC estimates based on 1,000 draws.N = 18656.Source: The authors.

Table 5 .
Scalar summary effects by hospital type.