Spatial dependence in hospital efficiency: a spatial econometric approach for Ecuadorian public hospitals

ABSTRACT This study analyses whether the efficiency of Ecuadorian public hospitals experiences spatial dependence. We investigate whether demand variations have affected public hospitals’ efficiency performance through direct and spillover effects since the adoption of the current constitution in 2008. Our analysis exploits an innovative two-stage approach to estimate hospital efficiency in the first stage and then applies a spatial econometric framework to disentangle direct and spillover effects in the second. The results confirm positive spatial interactions among public hospitals’ efficiency, as well as positive direct and spillover effects coming from demand increases, which have been reinforced since 2008.


INTRODUCTION
The importance of healthcare services around the globe is widely recognized. Investment in healthcare has risen significantly, as have healthcare costs as a proportion of gross domestic product (GDP); as a result, there is great policy emphasis on improving efficiency (Bloom et al., 2015).
The territorial assessment of healthcare services is a key aspect of this, as there may be many sources of geographical variation that can produce different health outcomes according to the area of study (Chandra & Staiger, 2007). Also, the recognition of significant geographical concentration for many health indicators has motivated an extensive use of spatial methods to analyse health economic issues (Moscone & Tosetti, 2014).
In developing countries, the efficiency of public hospitals' resource use is crucial given the pressing need for proper allocation due to their scarcity and limited healthcare budgets (Hafidz et al., 2018). However, the methods used to study healthcare efficiency in developing economies have shown little consideration for the spatial dimension as a catalyst for the effectiveness of selected determinants in shaping the degree of efficiency achieved by different healthcare providers. The importance of considering the spatial dimension when it comes the analysis of hospital efficiency is linked to the limited and uneven territorial access to healthcare that these economies present and how existing strategic interactions between hospitals may benefit these countries. 1 If hospitals behave strategically, then they will adapt to increases in the performance of their neighbours by adapting their own efficiency performance. This means thatwithin a bounded areapublic healthcare efficiency may increase without incurring significant public investment. In developing countries, with limited resources and budgets, this would entail saving public funding by means of tailored decisions and well-targeted funding. This way, an improved healthcare efficiency performance would translate into more healthcare access with the existing healthcare resources.
In this paper, we contribute to filling the existing gap in the literature on public healthcare efficiency for developing countries by focusing our analysis on hospital efficiency and apply it to the Ecuadorian context. 2 The Ecuadorian case represents a suitable context of analysis, since it is characterized by significant territorial disparities and spatial dependence that arise due to the existence of spillover effects (Mendieta Muñoz & Pontarollo, 2016;Szeles & Mendieta Muñoz, 2016).
These disparities have brought about significant technological heterogeneity between public healthcare institutions (Piedra-Peña & Prior, 2020), 3 in which the hospitals with higher technology are concentrated in the most developed municipalities. Hereinafter, we will refer to these municipalities as cantons. 4 The government of Rafael Correa carried out a series of political reforms (starting with the new constitution in 2008) that introduced many changes with respect to equal access to medical attention. The free healthcare provided by the Ministry of Health's hospitals, jointly with new social security and criminal code laws that made insurance coverage compulsory, are among the country's most salient policies (De Paepe et al., 2012). The new access to medical attention resulted in a higher inflow of patients to public hospitals (Orellana Bravo et al., 2017). 5 As a consequence, we can expect that the potential increase in demand has an effect on hospital efficiency in the short-run: a higher number of treated patients can lead to better use of hospital resources, which are usually well endowed but inefficiently exploited in developing economies (Hafidz et al., 2018). In other words, these hospitals have spare resources that are not used to provide medical treatment. The increase in the number of patients would force the hospital managers to make use of these unexploited resources and therefore hospital efficiency would also rise.
However, the increase (or decrease) of efficiency might not just affect a given hospital, but also those surrounding it, given that hospitals can have strategic interactions in terms of quality and efficiency (Longo et al., 2017) that are linked to the mobility of the demand. If new barriers of access to healthcare decrease, patients will seek treatment in (or be referred to) hospitals where they believe they will benefit from high-quality services. 6 In Ecuador, the criteria for the distribution of public funding for healthcare services are based on the healthcare needs and the size of the served population (Villacrés & Mena, 2017). Hence, this system generates incentives for hospitals to attract more patients. As a consequence, within a bounded area, surrounding hospitals can perceive how bigger hospitals are behaving and adapting to a changing reality and then can react by trying to capture some of this newly created demand by increasing their own quality (which will be constrained by their technological endowment). If the costs of providing more quality are increasing, then higher costs stemming from higher demand will reduce the incentives for cost control, thus reducing hospital efficiency. 7 Given that hospitals have to make a decision about their efficiency, they can also react by increasing or decreasing (strategic complements and strategic substitutes, respectively) their efficiency in reaction to the changes in the efficiency of neighbouring hospitals.
In light of this evidence, this study analyses whether public hospitals in the Ecuadorian healthcare system adapt their efficiency in response to changes in the efficiency of neighbouring hospitals. We tackle the question of whether demand variations affect the efficiency of public hospitals through direct and spillover effects, and whether that level of efficiency has significantly changed since the current constitution came into force in 2008. We make use of the hospital occupancy rate to measure demand. The occupancy rate has been widely used as an index to show the actual utilization of an inpatient health facility for a given time period, and is commonly applied in the literature to proxy medical resource utilization (e.g., Capps et al., 2010;Herwartz & Strumann, 2012Lindrooth et al., 2003;Mobley, 2003). Our research covers the period 2006-14 and uses hospital and cantonal data gathered from the public statistics of the Ecuadorian Institute of Statistics and Censuses (INEC) and the Ecuadorian Central Bank (BCE).
We contribute to the existing literature by generalizing the approach of Longo et al. (2017) by means of the nonparametric efficiency measurement analysis that accounts for both the panel structure of the data and the technological differences of the healthcare system developed by Piedra-Peña and Prior (2020) to obtain robust, time-varying efficiency scores. Thus, we can account for one efficiency measure that considers the use of multiple inputs to produce a given level of healthcare output, rather than relying on different productivity ratios that might produce mixed results. Also, we adopt a spatial panel econometric specification as a framework of analysis in order to disentangle direct and spillover effects that can affect the hospitals' efficiency performance.
Our main results identify a significant positive spatial dependence among hospitals in Ecuador, suggesting that their healthcare services are perceived as complementary in terms of efficiency. More specifically, we find that a 1% increase in the efficiency of neighbouring hospitals increases the efficiency of an observed hospital by 0.45%. Also, the higher demand for medical treatments reflects a positive association with efficiency, regardless of the technological group; in addition, this demand affects the efficiency of surrounding hospitals, providing evidence of spillover effects. We estimate that, on average, a 1% increase in a hospital's demand increases the efficiency of the same hospital by 0.13% and the efficiency of all neighbouring hospitals by 0.09%. Both direct and spillover effects have significantly increased since 2008. This result suggests that the reforms carried out after the constitution boosted the efficiency of the public healthcare system. From a policy perspective, our findings emphasize the importance of well-planned and targeted reforms and public funding. Decision-makers can focus on efficiencystrengthening reforms that regulate the input consumption of public hospitals to boost their performance, which in turn, through spillover effects, enhance the regional healthcare performance without increasing the allocation of resources or public investment. In this line, policy-makers should also be aware of the fact that new reforms that might translate (for example) into hospital congestion, which could have a detrimental effect not just on a particular hospital, but neighbouring hospitals as well. In this way, they can affect the entire regional healthcare system. The remainder of the paper is structured as follows. Section 2 presents a literature review. Section 3 introduces the theoretical framework as developed by Longo et al. (2017). Section 4 discusses the empirical strategy. Section 5 describes the dataset. Sections 6 and 7 present the estimation results and conclusions, respectively.

LITERATURE REVIEW
In order to implement our empirical framework, we combine two strands of literature that have not often been Spatial dependence in hospital efficiency: a spatial econometric approach for Ecuadorian public hospitals 921 exploited jointly. The first strand relates to the healthcare efficiency measurement literature (Cantor & Poh, 2018;Hafidz et al., 2018;Hollingsworth, 2008), which is based on production theory and relies on parametric and non-parametric methods to estimate efficiencies for healthcare institutions. 8 The estimated efficiencies are more reliable than single averages because they are consistent with the economic concept of Pareto efficient allocation, where those efficient units are either minimizing inputs or maximizing outputs in the production of health. The second strand tackles health economic issues in a framework of analysis where the spatial dimension plays a key role in shaping the behaviour of the economic agents. In this strand of literature there have been many applications related to different topics in health economics that address a spatial perspective; for a complete review of most of this literature, see Moscone and Tosetti (2014), Baltagi et al. (2018) and Tosetti et al. (2018).
Although spatial economic methods have been applied in much of the literature, a lack of empirical research addresses spatial dependence in healthcare efficiency analysis. A few and very recent papers address a joint study of healthcare efficiency analysis from a spatial perspective. Their empirical approach is mainly based on a two-stage strategy: estimating hospitals' efficiencies in a first stage, and applying spatial econometric models in the second stage to address for spatial autocorrelation in the data. For example, Herwartz and Strumann (2012) study whether the introduction of prospective hospital reimbursements based on diagnosis-related groups (DRG) caused an increase in the negative spatial autocorrelation of hospitals' efficiency due to competition for lowcost patients. They find a statistically significant presence of negative spatial autocorrelation among hospitals in Germany, which significantly increased after the financial reform. Herwartz and Strumann (2014) extend the analysis in Germany in order to identify efficiency gains as a consequence of the same financial reform. They fail to find any efficiency gains from the new incentive structure in Germany. Felder and Tauchmann (2013) also study the efficiency of healthcare provision in Germany and consider the spatial perspective. Their findings show that accounting for spatial dependence increases the estimated effects of federal states on district efficiency. Herwartz and Schley (2018) depart from these findings and consider socioeconomic characteristics that influence regional efficiency in the provision of healthcare services in Germany. They find that income, unemployment, the proportion of immigrants and educational level have an effect in shaping the efficient provision of healthcare services in German districts. Martini et al. (2014) analyse the trade-offs between hospital health outcomes (such as mortality) and efficiency using a ward-level set of hospitals in Lombardy, Italy. Their findings support the existence of a trade-off between mortality rates and efficiency, where more efficient hospitals have higher mortality rates but lower readmission rates. They also point out the role of the spatial dimension, since mortality rates are higher for hospitals subject to a high degree of horizontal competition, but lower for those hospitals having strong competition but high efficiency.
Most of the studies that refer to efficiency measurement and spatial structure have focused on developed countries. To our knowledge, only one paper provides evidence about the change in efficiency for health institutions in developing countries. Kinfu and Sawhney (2015) estimate the determinants of the efficiency of institutional delivery of maternal care in India. They exploit stochastic frontier analysis (SFA) models accounting for spatial interactions and heterogeneity in a one-step approach, finding substantial inefficiencies in maternal care services between and within states.
In this study, we contribute to this new emerging strand of literature and extend it by analysing a Latin American country: Ecuador. We depart from the framework proposed by Longo et al. (2017), but extend its application by taking into account robust efficiency measures over time. We also consider hospital technological heterogeneity in a spatial panel econometric approach in order to account for different effects on the efficiency derived from hospitals' technological restrictions. This methodology represents an important advance in applied literature that can eventually be extended to other countries sharing realities similar to that of Ecuador.

THEORETICAL FRAMEWORK
The building blocks of the theoretical model we refer to in this analysis were developed by Gravelle et al. (2014) and Longo et al. (2017). Their theoretical models consider strategic interactions in hospital quality and efficiency arising from spillover effects within a geographical area. The idea is that if hospitals compete within a given area, they will attract patients by increasing their quality. If neighbouring hospitals react by increasing (or decreasing) their own quality, then we identify that hospitals are strategic complements (substitutes) in their quality. Furthermore, the reduction in a hospital's demand that follows from an increase in their closest neighbour's quality also has an effect on its efficiency. The costs of increasing the quality of the treatment might also reduce incentives to control costs and thus reduce efficiency. With a limited hospital capacity and resources, higher quality (which might require longer length of stay, use of beds, etc.) may reduce cost containment effort, and hence hospital efficiency. In this way, hospitals can be strategic complements (or substitutes) in their efficiency and a neighbouring hospital's increase in efficiency can induce an increase or decrease in its own efficiency. For a detailed description of the model developed by Longo et al. (2017), see Appendix B in the supplemental data online.

EMPIRICAL STRATEGY
The first stage of our empirical strategy involves defining a measure of efficiency. We make use of the efficiency scores obtained in Piedra-Peña and Prior (2020). Their empirical strategy is mainly based on the panel data envelopment analysis (panel-data DEA) proposed by Surroca et al. (2016) and Pérez-López et al. (2018). As in classical DEA, the method is based on the concept of Pareto Efficiency Allocation, according to which a resource endowment is efficient when there is no other possible allocation that makes a decision-making unit (DMU) better off. 9 An efficient value would obtain a value of one and it can take an input or an output orientation. The former focuses on minimizing the input use, while the latter focuses on maximizing the output obtained in the production process (Piedra-Peña & Prior, 2020). The problem that arises with classical DEA is that it is a crosssectional approach, so for all time periods there will be a time-specific frontier and time-specific efficiency coefficients; therefore, each time period is analysed without any connection to the levels of activity of adjacent time periods.
To overcome this problem, Surroca et al. (2016) and Pérez-López et al. (2018) propose the so-called paneldata DEA. The advantage of panel-data DEA over other efficiency measurement analyses is that it allows one to estimate time-variant coefficients of efficiency considering the inherent panel data structure. In this way, we can obtain efficiencies for each year under evaluation. One of the principal advantages of this approach is that the results are robust to outliers and temporal random shock, and thus provides efficiency scores representative of the complete period.
Piedra-Peña and Prior (2020) extend this approach to account for technological heterogeneities of Ecuadorian public hospitals by applying multivariate techniques to obtain panel data-DEA efficiency scores for three different groups (clusters): high-tech, intermediate-tech and low-tech.
Thus, we take the efficiencies estimated in Piedra-Peña and Prior (2020) for every year under evaluation and use them in the first stage of our approach. In this paper, we follow an input-oriented efficiency measurement. We assume a variable return to scale (VRS) model to deal with heterogeneous observations. The efficiency frontier is developed by optimizing the weighted input/ output ratio of each DMU, subject to the condition that this ratio can be equal, but never exceed one for any other DMU in the data set. 10 Thus, conditioning the efficient hospitals to have scores equal to one and those inefficient ones to have values lower than 1.
The second step of our strategy defines a convenient spatial model whose main idea is to assess whether hospitals' efficiency is associated with the efficiency of nearby hospitals and with other observed and unobserved variables.
The identification of the source of spatial autocorrelation needs to be carried out in order to avoid model misspecifications and omitted variable bias. Following the strategy described in LeSage and Pace (2009) and Elhorst (2010), we begin with a spatial Durbin model (SDM) setting as a general specification and then test for alternatives. For the process of model selection, see Appendix E in the supplemental data online.
The model selection points to a spatial autoregressive combined (SAC) model as the appropriate framework of analysis. Thus, we specify the following spatial panel data SAC model estimated by quasi-maximum likelihood (QML): where the variable e it is the logarithm of the efficiency of the hospital i at time t, e jt is the logarithm of the efficiency of hospital i's neighbour ( j = i) at time t, and w ij are the spatial weights that capture the pattern of spatial dependence and the strength of potential interaction between units i and j. We control for the demand of hospital i in time t using the hospital occupancy rate, ocrate it . 11 The variable Z it is the vector including variables such as market share, mortality rate, and regional demographics, that can affect the efficiency of the hospital. The variable Ø captures the hospital fixed effects and g t is the time effect. Finally, 1 it is the error term. We define equation (1) in matrix form as: As for the specification of the components of the weight matrix W , we use two different specifications. The former (hereinafter W d ) is the inverse of the shortest Euclidean distance between any pair of spatial units (i and j), which has been commonly used in the literature when the data covers healthcare providers (Tosetti et al., 2018). The latter (hereinafter W v ) uses the inverse shortest time travel distance by car between any pair of locations (i and j), as in Gravelle et al. (2014).
The key parameters to be estimated for the spatial autocorrelation are the coefficients r and l. These measure the strength of the spatial dependence due to efficiency changes and to unobservable factors in neighbouring hospitals respectively, conditional on the vectors of explanatory variables. If r . 0, then a positive autocorrelation is found in the efficiency of hospital i and the efficiency of its neighbouring hospitals; the same is true for l. In other words, the parameter r measures the strategic interactions amongst hospitals, as captured by the neighbouring hospitals' efficiency (hence the endogenous effects). If a positive autocorrelation is found, then hospitals are behaving as strategic complements. Due to spillover effects, hospitals are increasing their own efficiency in reaction to their neighbours' efficiency increase. However, there may be other sources of spatial autocorrelation in our sample that are not properly captured in the model and these are measured by l. For example, hospital efficiency can be affected by the physicians' productivity: the concentration of these physicians in large hospitals, mostly located in developed cantons, can generate interactions Spatial dependence in hospital efficiency: a spatial econometric approach for Ecuadorian public hospitals 923 among them, giving rise to a spatial pattern that cannot be captured by the data. We estimate direct, indirect (spillover), and total effects by obtaining the matrix of partial derivatives of the expected values of e it , as proposed by LeSage and Pace (2009). So far, the literature on spatial healthcare economics has identified the existence of spatial spillovers based on coefficient estimates (Baltagi et al., 2018). We improve the empirical approach by accounting for the direct, indirect, and total effects of independent variables.
In addition, we carry out the LeSage and Pace (2009) partitioning analysis of the spatial multiplier. 12 With this, we are able to trace the effect of the linkages between demand levels of neighbouring hospitals. Thus, we also determine the impacts that the demand itself has over the higher order of contiguity.
Finally, to test the statistical variations of healthcare demand upon the hospitals' efficiency before and after 2008, we interact the logarithm of the occupancy rate with time dummies (ocrate t ). 13 Specifically, we have built the following test: H o : ocrate 1 = ocrate 2 , where ocrate 1 = 1/2 2008 t=2007 ocrate t , which represents the subperiod before the constitution, 14 while ocrate 2 = 1/6 2014 t=2009 ocrate t constitutes the subperiod after the constitution. These hypotheses are tested by means of a two-sided t-test. 15

DATA AND VARIABLES
The database we have used covers the period from 2006 (two years before the new constitution was approved) to 2014. The hospital information was collected from the Annual Survey of Hospital Beds and Discharges and the Survey of Health Activities and Resources provided by the INEC. We excluded psychiatric, dermatologic, and geriatric hospitals, and removed outliers from the sample. 16 We retrieved a panel data of 186 hospitals. Cantonal economic and demographic variables were retrieved from the BCE and INEC's public statistics respectively. For a description of all the variables, see Appendix D in the supplemental data online.

Variables for the efficiency measurement
As mentioned, we employed the efficiency estimations granted from Piedra-Peña and Prior (2020). The selection for both input and output variables was related to the existing literature on hospital efficiency measurement. For a complete overview, see Hollingsworth (2008), O'Neill et al. (2008) and Cantor and Poh (2018).
In our study, the input variables (controlled by the hospitals) are the number of beds, the medical equipment, and the availability of the infrastructure that is widely used as a proxy for hospital size and capital investment (O'Neill et al., 2008). To proxy for labour costs, clinical staff were usually included (Hollingsworth, 2003(Hollingsworth, , 2008. To this end, we included the number of physicians and healthcare professionals beyond the number of physicians of the hospital. To measure public hospitals' final production of health, the number of hospital discharges was employed. We controlled for the patients' case heterogeneity using the Herr (2008) case-mix index (see Appendix G in the supplemental data online).

Variables for the spatial econometric model
To account for the changes in the number of treated patients, we used the logarithm of the hospital occupancy rate. 17 Strumann (2012, 2014) point out the importance of this variable in relation to healthcare efficiency. It serves as a proxy to determine whether hospitals promptly adjust their working staff to the increase in treated patients. Thus, hospitals with a relatively low occupancy rate can be interpreted as having an oversized staff, and thus as being unlikely to meet the demand for patient care efficiently. 18 To provide a proxy for market structure in the hospitals' respective cantons, we used the logarithm of the hospital's market share. Market share has often been used as an explanatory variable in research regarding healthcare efficiency in developed economies to provide a measure of concentration (or competition). In our context, we additionally envisage two scenarios. In the first case, larger market shares could be related to larger hospitals, which are often located in more developed cantons. Piedra-Peña and Prior (2020) find that these types of hospitals are those with better technology and better performance (and are therefore the most efficient). The second case represent those hospitals located in less developed cantons (hence, with lower technology and efficiency) which do not have to deal with many close competitors.
We proxy hospitals' quality by including the logarithm of the hospital and cantonal mortality rates. Other morbidity variables were also included, such as the number of disease-specific treated patients, to provide additional controls on the complexity of the cases treated.
The technological differences were included as a dummy interacting with different hospital independent variables to estimate their differential effect on the hospitals' efficiency scores.
As for canton specific variables, we included the logarithm of the density, gross value added (GVA), and population over 65 years old.
Finally, we used the logarithm of cantonal patient migration measured as the number of patients treated in cantons different from the ones of their place of residence. Felder and Tauchmann (2013) state the importance of accounting for regional patient migration, as it can potentially be correlated with inefficiency. Patient migration can explain efficiency differences between territories, as it could be capturing deprivation effects (Herwartz & Schley, 2018). Bigger hospitals located in the developed regions are very likely to treat patients from outer regions, as patients in less developed regions have access restrictions to good healthcare quality and perceive these bigger hospitals as having higher quality than those located in their area of residence (Martini et al., 2014). In this way, smaller hospitalslikely to be located in less developed areascan present higher efficiencies that are not due to more efficient use of their inputs, but rather to a lower local demand due to patient migration (Herwartz & Schley, 2018). The descriptive statistics of our data are presented in Table 1. We split the sample in technology cluster according to the criterion proposed by Piedra-Peña and Prior (2020) (low-tech, intermediate-tech and high-tech).

Exploratory spatial data analysis (ESDA)
Before performing the more quantitative analysis, it is important to assess the true existence of spatial dependence in the distribution of the health resources in the Ecuadorian territory. Hence, we perform an ESDA by means of Moran's I-statistic to identify different patterns of spatial association and regional clusters or atypical locations of our observations and gain a better understanding of the spatial structure of the data. Figure 1 depicts the Moran's map and scatterplot for the mean of the number of physicians per hospital between 2006 and 2014. 19 We used the weight matrix W d based on the inverse Euclidean (shortest) distance between hospitals. 20 The evidence shown in Figure 1 shows an average positive spatial correlation for the hospital feature considered. Looking at the maps, the hospitals that present positive spatial autocorrelation (black points) are clustered around Quito and Guayaquil, which are the two biggest and most developed cantons in Ecuador (Mendieta Muñoz & Pontarollo, 2016). It is also worth noting that the spatial pattern changes as the distance of hospitals from these cantons increases. Hospitals that surround them present dissimilar amounts of resources (low-high), and present a negative correlation as they move farther away, as depicted by the greyish points (low-low).
This result assesses not only the high heterogeneity in terms of technological endowment for healthcare in Ecuador, but also the uneven distribution of these high-tech hospitals within the territory. Table 2 shows the regression results from the SAC spatial econometric model for equation (2). 21 The first set of estimations refers to the model with the selected weight matrices and without incorporating the technological discrepancies. Hereinafter, we label this first type of setting as the baseline model. 22 The results confirm the existence of positive spatial dependence among hospital efficiencies in the sample. These results are robust for both types of spatial matrices. Considering the weight matrix based on the shortest travel time distance, 23 the estimate of r indicates that a 1% increase in the efficiency of neighbouring hospitals j increases the efficiency of the hospital i by 0.45%. Referring to our efficiency measurement setting, the results suggest significant strategic complementarity effects in hospitals' efficiency. This evidence suggests the existence of competition effects among hospitals in terms of efficiency due to strategic interactions within hospitals in a bounded area. Hospitals react to changes in the efficiency of neighbouring hospitals by adapting their own efficiency due to existing spillover effects.

ESTIMATION RESULTS
The statistical significance of the estimates for l suggests the presence of a negative spatial error correlation. This result involves the existence of other sources of spatial correlation in our sample that were not properly captured in the model. The results are in line with previous findings in the literature. The existence of spatial error correlation is not new in spatial health econometrics (Baltagi et al., 2018). There are several risk factors that are difficult to measure, but they are so geographically concentrated that they affect health outcomes (Tosetti et al., 2018).
Due to the scarce literature that exploits a similar approach, especially for Latin American countries, a comparative analysis becomes difficult. Nevertheless, the sign of the spatial correlation and the effect of both the spatially lagged efficiency score and the error term are in line with those of Felder and Tauchmann (2013). Although they perform a cross-sectional analysis at the district level in Germany, the average effect of spatial dependence on the hospitals' efficiencymeasured by efficiency measurement non-parametric modelsdoes not seem to be unrealistic in the Ecuadorian context. Table 2 provides additional information about spillover effects. We present total effects disaggregated in direct and indirect (spillover) effects (LeSage & Pace, 2009). The logarithm of occupancy rate shows that an increase in 1% in a hospital's occupancy rate increases the efficiency of the same hospital by 0.13% and the efficiency of all neighbouring hospitals by 0.09%. 24 This finding is in line with the argument of the inefficient use of spare resources in the public healthcare system as argued by Herwartz and Strumann (2014).
Instead, market share is associated with a negative estimated coefficient. Its direct and indirect effects show that a 1% increase in this variable diminishes efficiency performance by 0.03% for the selected hospital and 0.02% for neighbouring hospitals. This implies that hospitals that host more patients tend to experience an inefficient use of resources. However, the magnitude of this effect could be different in accordance with the type of hospital under consideration.
We go a step further than the applied literature by including technological differences as interactions with hospital-related variables, since these are the ones that tend to be relevant for the analysis (Piedra-Peña & Prior, 2020). Table 3 presents the estimated results including technological interactions. Model (1) presents the baseline model using W v . Models (2-4) show the estimation results with the covariates at the hospital level interacted with two dummies of cluster 2 (intermediate-tech) and cluster 3 (high-tech). 25 The most interesting finding refers to the market share. The estimated coefficient is significant and robust, positively associated with the technological endowment Spatial dependence in hospital efficiency: a spatial econometric approach for Ecuadorian public hospitals of public hospitals: the estimates are positive for intermediate-and high-tech hospitals, something which is at odds with previous results. Indeed, the estimations provide evidence that, in case of more concentration, intermediate-and high-tech hospitals' efficiency performance increases, enforcing spillover effects. These results are not far from recent findings in the literature. Pross et al. (2018) find that concentration at the regional and hospital level can improve quality and resource efficiency. Gobillon and Milcent (2013) identify that the higher local concentration of patients in a few large hospitals rather than many small ones improves the hospitals' performance. As these authors suggest, this could be the result of a learning-bydoing process. The hospitals with the best technology, having treated more patients and more severe cases over time, experience improvements in their treatment capacity through experience. Our estimations also stress that there are no significant changes in occupancy rate and mortality rate when referring to the technological endowment. Regardless of technological differences, higher demand translates into higher efficiency. The reason for this might be that all hospitals, regardless of their technological level, show low levels of efficiency, implying an inefficient use of their spare inputs, which gives them room for improvement when there is a higher demand for medical treatment.
To discern how the occupancy rate has influenced the efficiency of public hospitals, in Figure 2 we plot the tendency of the total effect of ocrate t over time. There is a notable cut in the total effect after 2008, suggesting that the increase in demand after this year yields an increase in hospital performance due to a more efficient use of spare resources. This effect might also be the result of proper managerial planning that could have anticipated an increase in the bulk of patients, given that the Ecuadorian population had time to become informed about the potential changes that the constitution embraced. Spatial dependence in hospital efficiency: a spatial econometric approach for Ecuadorian public hospitals 927 To verify whether this discontinuity in efficiency was statistically significant, Table 4 presents the correspondent hypotheses tests for both direct and indirect effects. The test rejects both hypotheses with 95% confidence. This result implies that the period after the adoption of the new constitution enforced not only a significant upturn in the direct effect that an increase in demand generated in a specific hospital, but also bigger spillover effects for neighbouring hospitals as well.
The results presented so far highlight the importance that covariates (mainly higher demand and more competition) can bring to the efficiency performance of the public healthcare system, and the potential effect that policy implementation can have on it when it is well planned at the territorial level. As has been proved, these policies do not bring benefits exclusively for the selected hospital, but also affect neighbouring hospitals due to spillover. Nevertheless, it is worth pointing out that there are still some explanatory variables that worsen the performance of the system. Some of these are still unknown and more research must be done in this direction.
Finally, Table 5 presents the different neighbouring order coefficient estimates of the partitioning analysis. The direct partitioning effect in Table 5 shows a significant impact beyond the so-called zero-order neighbour (W 0 ) (see Appendix F in the supplemental data online) that decreases significantly in size from W 1 on. This implies that for direct impacts, those immediate neighbours play a strong role. 26 Regarding the indirect partitioning effects, these are significant for the second-, third-and fourth-order neighbours (and significant at 90% confidence for the fifth-order neighbour, considering W v ) and strongly decreasing in size after W 3 . This effect suggests that, although significant, demand has a limited effect over space for hospital efficiency, with spillover effects being strong in small, concentrated areas and generating small feedback effects.

CONCLUSIONS
This study analyses the spatial dependence of hospital efficiency in Ecuador. To address this question, we apply an innovative methodology proposed by Piedra-Peña and Prior (2020) to obtain robust efficiency scores for a sample of public hospitals in Ecuador between 2006 and 2014, taking into account their technological differences to avoid biased results. We then use this efficiency score as a dependent variable in a spatial econometric SAC model to consider spatial autocorrelation in efficiency and disturbances. The results confirm that an increase in the efficiency of surrounding hospitals increases the efficiency of a selected hospital. The direction of these effects is robust to different specifications and estimation methods. 27 Spatial autocorrelation and spillover effects seem to diminish as the hospitals' distance from the most developed areas increases. As Longo et al. (2017) state, the positive dependence between neighbouring hospitals suggests that they are acting as strategic complements in efficiency.
We also address the question of whether the variations in demand for a given hospital, which we measure through occupancy rates, affect nearby hospitals' efficiency through spillover effects. The results confirm that increases in demand for medical services for public hospitals cause neighbouring hospitals to attract some of this demand, and that this boosts their own efficiency, regardless of the technological endowment of the hospitals. A large portion of this positive effect can be explained by the low levels of occupancy rates that the public healthcare hospitals show, which might imply the existence of spare     Spatial dependence in hospital efficiency: a spatial econometric approach for Ecuadorian public hospitals resources that are inefficiently used to produce healthcare outputs. The increase in demand forces hospitals to make better use of these spare resources, and so their efficiency performance rises. In addition, the estimates show that after 2008, the direct and indirect impact of occupancy rates on the efficiency performance significantly increased. While waiting for the approval of the new constitution, which was expected to entail an increase in the number of patients seeking medical treatments, hospital managers could have planned strategies to adapt to these changes, and this could in part be reflected in this higher effect after 2008. The technological disparities among hospitals also play a key role, especially when analysed jointly with market share. We find evidence that intermediate-and hightech hospitals have a differential effect. That is, the increase in concentration of patients in technologically better hospitals increases their efficiency (as well as that of surrounding hospitals), whereas the opposite effect is found for low-tech hospitals. These results provide some evidence of a potential learning-by-doing process in intermediate-and high-tech hospitals.
These differences have important policy implications. Given that high-tech hospitals are mostly concentrated in developed areas, policy decisions and public funding should be allocated taking into consideration the territorial development within the country. The rationale is that policy reforms and public investment that imply more competition (due to the construction of more hospitals) can be counterproductive for the healthcare performance of developed areas but beneficial for less-developed ones.
In this line, policymakers could exploit spillover effects in developed areas to reinforce hospital performance. However, they should be aware that these spillover effects will spread to a limited extent over space, which emphasizes the importance of well-targeted policy decisions. Clearer criteria for public funding allocation and stronger regulation of hospital resource consumption that control (or limit) for inflation of   existing hospitals to compete for patient inflow by increasing their quality and efficiency. These improvements could be a potential solution that could reduce the existing regional gap in the Ecuadorian healthcare system. Finally, we need to point out some issues referring to data availability. It is recommended that future research take into account further information that has been proven to have a significant effect on hospitals' efficiency, such as the quality of treatments and budgetary information.
Future research should fill this gap to drive additional empirical research in order to bring relevant insights for policy decisions. In this line, a clear suggestion for policymakers is to implement strong monitoring systems that provide researchers and healthcare managers with reliable and robust data. the scientific committee of the PhD in Applied Economics, Nicola Pontarollo, Judit Vall, and the anonymous referees for their valuable comments. Any remaining errors are my own responsibility.

DISCLOSURE STATEMENT
No potential conflict of interest was reported by the author.

NOTES
1. The term 'strategic interactions' is used in the literature to refer to the interdependence among features or actions of selected units stemming from competition between those units. Strategic interactions arise due to the existence of spillover effects (Brueckner, 2003) that cause the levels of the variables of one unit to be affected by the levels of the same variables of neighbouring units. 2. See Appendix A in the supplemental data online for a description of the institutional healthcare setting in Ecuador. 3. Here we consider technology as the set of constraints defining how one can combine or convert inputs (e.g., number of physicians, beds, etc.) into outputs (e.g., number of discharges or procedures carried out) in the production process. In this particular context, this can relate to the availability of human capital, infrastructure, etc. 4. In 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 in turn are subdivided into parishes. 5. According to the Public Ministry of Health (MSP), between 2006 and 2010 the number of surgeries increased by 47% and hospital discharges by 43% (Ministerio de Salud Pública, 2012).
6. As will be described in following sections, Ecuador has a reduced number of well-endowed public hospitals that are capable of treating complex diseases where patients can receive free medical treatment due to the reforms carried out since 2008. So, given the reduced barriers of access, it is very likely that patients perceive these hospitals as providing the best quality they can obtain. In fact, Piedra-Peña (2021) demonstrates that those best-performing and well-endowed hospitals, located in developed regions, have a significant pulling effect to attract patients. These movements might also be induced by hospital reputation, built over years, referrals or public news. 7. In fact, according to Villacrés and Mena (2017), the current funding scheme of the country can generate inefficiencies given that hospitals have an incentive to attract patients and inflate costs. 8. Although the scope of our study is in line with nonparametric methods, see Colombi et al. (2017) for a thorough discussion of the recent literature using parametric stochastic frontier models. 9. We can apply DMU to any unit of analysis such as individuals, departments, firms, municipalities, or, in the case of this study, hospitals. 10. For an in-depth description of Piedra-Peña and Prior's (2020) approach, see Appendix C in the supplemental data online. 11. For a discussion on how occupancy rate is used to proxy hospital demand, see section 5.2. 12. See Appendix F in the supplemental data online for an explanation of LeSage and Pace's (2009) spatial effects and the respective partitioning analysis. 13. We make use of hospital occupancy rate to measure demand. This is further explained in section 5.2. 14. The constitution came into force in October 2008. 15. Regarding our two-stage application, some remarks are in order. As pointed out by Simar and Wilson (2007), conventional regression models (such as ordinary least squares -OLS) applied in the second stage yield biased results because the efficiency scores estimated in the first stage are serially correlated. In addition, possible correlation of the contextual variables with the error term is another possible source of bias. The authors propose new methods based on bootstrapping to overcome these problems. We therefore employ maximum likelihood (ML) estimates that are consistent with involving DEA efficiency scores. Plus, the logarithmic transformation of the efficiency scores ensures an unbounded dependent variable and thus enables a consistent ML estimation (Simar & Wilson, 2007). Finally, we ensure valid inference of the estimated marginal effects by simulating the distribution of the direct and spillover effects using the variance-covariance matrix implied by the ML estimates, as proposed by LeSage and Pace (2009). 16. We excluded psychiatric, dermatologic and geriatric hospitals because they focus on specific illnesses and patients that require different treatments that could bias the efficiencies. 17. All the variables expressed as percentages were on a 0-100 scale before obtaining the logarithms in order to facilitate the estimations and interpretation of the results.
18. We acknowledge that occupancy rates may also increase either by reducing the number of beds or by augmenting the patient's length of stay, and not just due to the increase in hospital admissions. However, in our dataset, neither the average number of beds nor the number of bed days per patient has drastically changed over the years. The former went from 75 beds in 2006 to 86 in 2014, on average. The latter went from 8.8 days of care per patient in 2006 to 11.67 in 2014. Hence, it is valid to attribute the sudden change in occupancy rate in our data (as will be seen in the following sections) to the increase in demand for medical care. 19. We use hospital feature averages in this analysis given that Moran's I-statistic is a cross-sectional approach. We also carried out the same analysis for each year. The results are comparable and available from the author upon request. 20. We reproduced the same test for three other hospital features: number of beds, medical equipment and hospital personnel (not including physicians). The results are comparable and available from the author upon request. 21. Complete results tables are available from the author upon request. 22. We also estimate progressive estimations to check the robustness of our results to the inclusion of control variables. The results are robust and the size is comparable. The estimations are shown in Appendix H in the supplemental data online. 23. Henceforth, this will be used for interpretation because it is a more realistic matrix of hospital interactions than that of Euclidean distances (W d ). 24. We tested the direction of the causality between hospital efficiency and demand as well as for hospital efficiency and market shares. We used Granger's (1969) causality test for panel data models as adapted by Dumitrescu and Hurlin (2012). The test rejects the null hypothesis of non-causality in both cases. 25. Estimation results with the other weight matrix are comparable and available from the author upon request. 26. The impact of the marginal change in demand for hospital i on its own efficiency is the result of local effects plus feedback effects that pass mainly through its direct neighbour j. 27. Robustness analysis is carried out in Appendix H in the supplemental data online.