CHAPTER 2: REVIEW OF PRODUCTIVITY AND EFFICIENCY ANALYSIS
2.3 Measuring efficiency
2.3.2. Nonparametric methods
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all producers (Alvarez et al., 2012). This assumption may not be appropriate since firms in a particular industry may use different technologies (technological heterogeneity), in which case the estimated underlying technology is likely to be biased (Orea & Kumbhakar, 2004; Alvarez et al., 2012). Failure to account for these unobserved technological differences during estimation may result in them being incorrectly labelled as inefficiency (Orea & Kumbhakar, 2004).
Production heterogeneity can be addressed through the use of either two-stage or one-stage methods. The two-stage approach involves first separating the sample into several groups, based upon some a priori sample separation information, and then conducting separate analyses for each group (Orea & Kumbhakar, 2004). The one-stage method is an attractive alternative with the ability to separate the sample into groups and estimate the technology for each of these groups in one step (Alvarez & del Corral, 2010). A comparison of the one-stage approach, commonly referred to as a latent class (mixture) model, and the two-stage approach found the latent class model to be a superior method (Alvarez et al., 2012). By incorporating the latent class model (LCM) into the stochastic frontier framework, Orea & Kumbhakar (2004), Alvarez & del Corral (2010) and Alvarez et al. (2012) were able to estimate efficiency, while accounting for technological heterogeneity.
Tsionas & Kumbhakar (2004) indicated that the latent class model may not be realistic in certain cases, drawing attention to the fact that there may be some persistence in the movement from one group to another and the lack of parsimony of the model. As a potential remedy, Tsionas &
Kumbhakar (2004) proposed a stochastic frontier model with a Markov switching structure in which parameters were allowed to take a finite number of possible values, and at each time period there was a probability that the parameter values will remain unchanged or switch to something different. This method has the advantage of considering both cross-sectional and temporal heterogeneity, something the previous LCM’s failed to achieve.
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and one, with zero representing the lowest efficiency measure and one representing optimum efficiency (equal to that of the best-practice firm) (Stokes et al., 2007; Delis et al., 2009).
Efficiency scores lower than one indicate that the same vector of outputs could be produced with a smaller vector of input, therefore reflecting the presence of inefficiencies in production (Andersen & Petersen, 1993).
Nonparametric frontier methods originated from the work of Farrell (1957) which involved the use of linear programming techniques to construct a free disposal convex hull of the observed input-output ratios (Førsund et al.,1980). This approach was extended by Charnes et al. (1978) who adopted a mathematical programming approach to efficiency analysis which is commonly known as Data Envelopment Analysis (DEA). Central to the DEA approach is the assumption of convexity of the production possibilities set (Delis et al., 2009). The Free Disposal Hull (FDH) method is an extension of DEA that allows for nonconvex production possibility sets by assuming free disposability of inputs and outputs (Simar & Wilson, 1998; Delis et al., 2009). Both approaches allow efficiency to vary over time.
DEA has the advantage of not requiring the specification of a production technology or distributional assumptions regarding the error term (Sharma et al., 1999). Furthermore, it allows for the simultaneous use of multiple inputs and multiple outputs, each being measured with different units of measurement (Wadud & White, 2000). It is, however, criticized for its deterministic nature, attributing all deviation from the frontier to inefficiency. As a result, DEA is likely to be highly sensitive to measurement error and statistical noise (Sharma et al., 1999).
DEA models may estimate efficiency with either input or output orientations (Stokes et al., 2007;
Murova & Chidmi, 2011). Input-oriented models measure technical inefficiency as a proportional reduction in input usage, holding output levels constant. Output-oriented models measure technical inefficiency as a proportional increase in output production, holding input levels constant (Coelli et al., 2005). These two orientations provide equal estimates under the assumption of constant returns to scale (CRS) but not for variable returns to scale (VRS) (Delis et al., 2009). There is a lack of consensus among the literature as to which orientation is the “best choice” (Delis et al., 2009). Coelli et al. (2005) note that the choice of orientation depends upon the nature of the industry and should be selected according to which quantities (input or output) the firm has the greatest control over.
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Early DEA models such as that of Charnes et al. (1978) assume CRS, which permits the estimation of an “overall” measure of technical efficiency. The CRS assumption that all firms operate at an optimal scale may not be appropriate since, in reality, a number factors such as imperfect competition, government regulations and financial constraints cause a firm to operate at a non- optimal scale (Coelli et al., 2005). Banker et al. (1984) extended the work of Charnes et al. (1978) to consider VRS, which permitted the separation of overall technical efficiency into pure technical efficiency and scale efficiency components. Furthermore, overall technical efficiency was found to be equal to the product of pure technical efficiency and scale efficiency. Analysis with the assumption of VRS is considered more flexible and envelopes the data in a tighter manner than CRS (Sharma et al., 1999). One deficiency of the VRS measure of scale efficiency is that it fails to indicate whether the firm is operating under increasing or decreasing returns to scale (Coelli et al., 2005). This can be determined by solving a non-increasing returns to scale (NIRS) DEA model (Sharma et al., 1999). If the technical efficiency (TE) measure under NIRS is equal to that under CRS, there are increasing returns to scale. However, if the TE measure under CRS is less than that under NIRS, there are decreasing returns to scale (F𝑎̈re et al., 1994, as cited by Sharma et al., 1999)
Simar & Wilson (1998) introduced the bootstrap method as potential tool to analyse the sensitivity of measured efficiency scores to the sampling variation of the estimated frontier. The bootstrap method is based upon the idea of repeatedly simulating the data generating process (DGP), through resampling, and applying the original estimator to each of the simulated samples so that the resulting estimates mimic the original estimator’s sampling distribution (Simar & Wilson, 1998).
This allows researchers to conduct traditional hypothesis tests and construct confidence intervals (Coelli et al., 2005). Simulating the DGP, however, can prove difficult since the bootstrap method requires that a clearly defined model of the DGP is known, otherwise it is not possible to determine whether the bootstrap accurately mimics the sampling distribution of the original estimators (Simar
& Wilson, 1998).
Two-stage DEA represents an attempt to simultaneously estimate farm level efficiency and explain the reasons for the resulting estimates of efficiency. This method involves estimation of the efficient frontier and firm level efficiency scores in the first stage (a conventional one-stage DEA).
In the second stage, these efficiency estimates are regressed against a set of explanatory variables
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in an attempt to explain observed inefficiency (Balcombe et al., 2008; Johnson & Kuosmanen, 2012). Despite several applications in the literature (Wadud & White, 2000; Helfand & Levine, 2004), the two-stage DEA approach has fallen under criticism due to several key limitations.
Firstly, studies which have applied the method are criticized for failing to describe the underlying DGP, therefore raising doubt as to the meaning of the estimates. Secondly, two-stage DEA estimates have been found to be serially correlated and as a result standard approaches to statistical inference are invalid (Simar & Wilson, 2007). Simar & Wilson (2007) proposed an application of the double bootstrap method to DEA as a means of overcoming these limitations.
In an attempt to circumvent the limitations of parametric stochastic frontier models, without foregoing their advantages, Kumbhakar et al. (2007) proposed a nonparametric stochastic frontier model based on the local maximum likelihood procedure (LML). This method adopts local modelling techniques which do not require strong assumptions regarding functional form and differ from traditional nonparametric approaches, such as DEA, in the sense they are able to provide efficiency estimates that account for random noise (Serra & Goodwin, 2009; Guesmi et al., 2013). Furthermore, local modelling techniques can accommodate heterogeneity in the data by making the variances of both components of the error term observation specific (Serra & Goodwin, 2009). Due to the complexity involved in its implementation, this approach has received limited application in empirical studies (Guesmi et al., 2013).
Dai (2016) proposed a fully nonparametric, three-stage method of efficiency estimation using the Richardson-Lucy blind deconvolution algorithm (RLb) to decompose firm specific inefficiency from their composite errors. In the first stage, the shape of the frontier is estimated using convex nonparametric least squares (CNLS) regression and the residuals are estimated. In the second stage, the expected inefficiency for all firms is estimated and used to correct the CNLS residuals estimated in stage one. Finally, stage three involves the estimation of firm specific efficiencies using RLb. This model does not require any distributional assumptions, is insensitive to statistical noise in the data and is robust to heteroscedasticity. Despite its potential advantages, RLb is sensitive to frontier estimation (the difference between the estimated and true frontier) and may be biased and thus should be applied with caution.
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