Title: Unobserved heterogeneity in stochastic cost frontier models: an application to Swiss nursing homes
Abstract: Abstract This paper applies a number of stochastic cost frontier models to a panel data set and compares their ability to distinguish unobserved heterogeneity from inefficiency variation among firms. The main focus is on Greene's Citation2005 panel data model that incorporates firm-specific effects in a stochastic frontier framework. In cases where the unobserved heterogeneity is correlated with explanatory variables, while the random effects estimators can be biased the fixed effects model may overestimate inefficiency. In line with Mundlak, a simple method is proposed to include such correlations in random effects specification. The sample includes 36 Swiss nursing homes operating from 1993 to 2001. The results suggest that the proposed specification can avoid the inconsistency problem while keeping the inefficiency estimates unaffected. Acknowledgements The authors are grateful to the Ticino's Department of Health and Social Services for providing the data and their general support. Ilaria Mosca, Chiara Gulfi and Giorgio Borradori provided an excellent assistance in preparing the final data set. An earlier version of this paper was presented at the 8th European Workshop on Efficiency and Productivity Analysis, whose participants are thanked for their insightful discussion. The authors also wish to thank William Greene, Subal Kumbhakar, Robin Sickles, Luca Crivelli, the editor and an anonymous referee for their helpful suggestions. Any remaining errors are solely the responsibility of the authors. Notes 1 Note that most of the panel data used in the literature cover periods from 5 to 10 years. 2 Pitt and Lee's (Citation1981) model is different from the conventional RE model in that the individual-specific effects are assumed to follow a half-normal distribution. Important variations of this model were presented by Schmidt and Sickles (Citation1984) who relaxed the distribution assumption and used the GLS estimator, and by Battese and Coelli (Citation1988) who assumed a truncated normal distribution. 3 See also Heshmati and Kumbhakar (Citation1994) and Kumbhakar and Hjalmarsson (Citation1995) for two applications of this model. Note that in the latter paper, it is assumed that both time- and firm- specific effects are part of inefficiency. 4 The term 'heterogeneity bias' is used by Chamberlain (Citation1982) to refer to the bias induced by the correlation between individual effects and explanatory variables in a general RE model. 5 Notice that this model is different from those authors' other model discussed earlier. 6 This argument is based on an analogy with a GLS model that can be transformed to a 'within' estimator by using Mundlak's specification. However, it should be noted that given that the residual term in frontier models is asymmetric it is not clear whether this modification has the same effect in these models. 7 In Switzerland, in addition to the usual nursing care, nursing homes also provide basic medical services to their residents. 8 For a similar cost model specification see Filippini (Citation2001). 9 These guidelines are only recommendations and the nursing homes are not required to exactly follow them. 10 See Cohen and Spector (Citation1996) and McKay (Citation1988) for a similar approach in cost model specification for nursing homes. Cohen and Spector measured quality of care by staff to resident ratios. McKay used 'nursing hours per patient' to measure the nursing home's quality. 11 For more details on the functional form of the cost function see Cornes (Citation1992), p. 106. 12 Translog functional form requires that the underlying cost function be approximated around a specific point. In our case this point is taken as the sample median. We choose the median rather than the mean, because it is less affected by outliers and thus the translog approximation will have a better precision. 13 Switzerland's global consumer price index has been used. However, since total costs and prices are normalized by the capital price this adjustment is not necessary for the regressions. 14 See Bös (Citation1986), p. 343. 15 See Baltagi (Citation2001) for an extensive discussion. 16 Note that here the cost-efficiency does not include scale efficiency. 17 This is a practical issue rather than a modeling problem. In fact the FE model is more general in that it does not assume a single underlying population for all the firms. 18 See also Hsiao (Citation2003), pp. 44–46, for a proof of the identity of Mundlak's GLS estimators and FE estimators. 19 This model is a special case of a stochastic frontier model with random parameters (in this case random intercept). 20 The average dependency index, which is included in the model, only measures the time required for nursing care, thus captures only one aspect of case-mix severity. Other factors like the need for medical treatment and drugs are not observed. 21 It is worth noting that here cost inefficiency is defined as the excess costs due to the firm's technical problems or to suboptimal allocation of resources. Other inefficiency sources like scale inefficiencies, which are beyond the firm's control are excluded. 22 As our specification does not include any time-invariant factor, this statement does not apply here. 23 In distributions such as half-normal or exponential, perfect efficiency is at the mode of the probability distribution, thus the most likely outcome. 24 The only exception is the FE models that interpret the effects as inefficiency. 25 There are some nursing homes that offer the possibility of nursing care in external residential apartments. The nursing care of this type is less intensive (thus less costly) than the care given to the home's residents. For this reason we excluded four nursing homes whose share of external beds is more than 10% of their total beds. In our final sample there remain two nursing homes that offered external care (less than 10%) for some years during the study period. 26 A more precise estimation of capital stock would requires capital inventory data, which are not available to us. 27 These findings are in line with the results obtained by Filippini (Citation2001) using a shorter panel and a slightly different number. of nursing homes. 28 Our results indicate that the Hessian matrix of the estimated cost function with respect to input prices (labour and capital) is not negative semi-definite, thus the concavity condition is not satisfied in any of the specifications. 29 See Farsi and Filippini (Citation2004) for a more detailed discussion. 30 These models have also the highest correlation coefficients in efficiency ranks (0.98 between 2 and 5, and 0.95 between 3 and 6). 31 The expression 'mutual consistency' is used by Bauer et al. (Citation1998) in this context. 32 Note that each year subgroup has about 36 observations. 33 Horrace and Schmidt (Citation1996) show that a panel with six periods cannot provide a consistent estimation of individual efficiency scores.
Publication Year: 2005
Publication Date: 2005-10-10
Language: en
Type: article
Indexed In: ['crossref']
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Cited By Count: 123
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