2.1.1 The input distance function

Following Shephard (Reference Shephard1953), the input distance function (IDF) can be defined in relation to the input set $L\left ( y\right )$ or the production technology $T$ as follows:

\begin{equation*} D_{I}\left ( y,x\right ) =\max _{\vartheta _{I}}\left \{ \vartheta _{I}:\ \frac {x}{\vartheta _{I}}\in L\left ( y\right ) \right \} =\max _{\vartheta _{I} }\left \{ \vartheta _{I}:\ \left ( \frac {x}{\vartheta _{I}},y\right ) \in T\right \} \end{equation*}

where $1/\vartheta _{I}$ represents the proportional contraction of inputs required to reach the inner boundary of the input set, or the production frontier, with the outputs held constant. The function $D_{I}\left (y,x\right )$ is defined as the ratio of the observed input to the minimum input required to produce the given output. Therefore, for any input vector $x$ , the expression $x/D_{I}\left ( y,x\right )$ represents the minimum input vector on the ray from the origin through $x$ that can produce $y$ , as illustrated in Figure 1. Efficient producers, who produce on the boundary of the input set, or the production frontier, have $D_{I}\left (y,x\right ) =1$ . Inefficiency is indicated by $D_{I}\left ( y,x\right ) \gt 1$ .

The Debreu-Farrell input-oriented measure of technical efficiency is defined as follows:

\begin{equation*} TE_{I}\left ( y,x\right ) =\underset {\vartheta _{FI}}{\min }\left \{ \vartheta _{FI}:\ \vartheta _{FI}x\in L\left ( y\right ) \right \} =\underset {\vartheta _{FI}}{\min }\left \{ \vartheta _{FI}:\left ( \vartheta _{FI}x,y\right ) \in T\right \} \end{equation*}

The Debreu-Farrell input-oriented measure of technical efficiency is defined as the reciprocal of the $IDF$ : $TE_{I}\left ( y,x\right ) =\left [ D_{I}\left (y,x\right ) \right ] ^{-1}$ . The expression $TE_{I}\left ( y,x\right ) \leq 1$ represents a radial reduction of inputs required to be considered as being efficient. Technical inefficiency is defined as follows:

\begin{equation*} TI_{I}\left ( y,x\right ) =1-TE_{I}\left ( y,x\right ) =1-\frac {1} {D_{I}\left ( y,x\right ) } \end{equation*}

where $0\leq TI_{I}\left ( y,x\right ) \leq 1$ . The input distance function has the following properties [see Färe and Primont (Reference Färe and Primont1995), and Färe and Grosskopf (Reference Färe and Grosskopf2004) for more details]

1. i) representation, $D_{I}\left ( y,x\right ) \geq 1$ iff $x$ $\in L\left ( y\right )$ or $\left ( x,y\right ) \in T$

2. ii) non-increasing and quasi-concave in outputs, and

3. iii) non-decreasing, concave, and linearly homogeneous in inputs, $D_{I}\left ( y,\lambda x\right ) =\lambda D_{I}\left ( y,x\right )$ , $\lambda \gt 0$ .

The input distance function has been applied across various sectors to assess efficiency. In the manufacturing sector, it has been used to evaluate the efficiency of production processes, as shown by Hattori (Reference Hattori2002) and Atkinson et al. (Reference Atkinson, Färe and Primont2003). In the banking sector, it has been utilized to measure the efficiency of banks by examining the use of inputs, such as capital, labor, and other resources, in providing financial services and identifying strategies for performance enhancement, as discussed by Sturm and Williams (Reference Sturm and Williams2008). Additionally, the IDF has been applied to evaluate the efficiency of agricultural production by analyzing how inputs are transformed into outputs and identifying areas for improvement, as demonstrated by Tsionas et al. (Reference Tsionas, Kumbhakar and Malikov2015). Furthermore, the IDF has been utilized to address undesirable outputs. This is modeled by holding desirable outputs $y$ constant and treating undesirable outputs, denoted as $b$ , as inputs $x$ ; $D_{I}\left ( y,x,b\right ) =\max _{\vartheta _{I}}\left \{ \vartheta _{I}: \left(\frac {x}{\vartheta _{I}},\frac {b}{\vartheta _{I}}\right)\in L\left ( y\right ) \right \} =\max _{\vartheta _{I} }\left \{ \vartheta _{I}:\left ( \frac {x}{\vartheta _{I}},\frac {b}{\vartheta _{I}},y\right ) \in T\right \}$ . See, for example, Atkinson and Dorfman (Reference Atkinson and Dorfman2005). This approach credits producers for proportionally reducing both inputs and undesirable outputs to reach the production frontier. However, if inputs are freely disposable, undesirable outputs are as well. This treatment of undesirable outputs has been criticized due to the implied strong disposability of undesirable outputs, as noted by Färe et al. (Reference Färe, Grosskopf, Noh and Weber2005). Similarly, treating undesirable outputs as inputs allows for substitutability or complementarity among them. Furthermore, these studies overlook the fact that the production of undesirable outputs is influenced by the production of desirable outputs, represented as $b=f(y)$ , rather than the opposite, $y=f(b)$ . Thus, treating undesirable outputs as inputs is inappropriate because it imposes incorrect theoretical restrictions on the production technology. Therefore, the IDF can be used only to decrease inputs while keeping both outputs constant. Assaf et al. (Reference Assaf, Matousek and Tsionas2013) utilized the IDF and treat undesirable output as a technology shifter.

Figure 1. The Debreu-Farrell input-oriented measure of technical efficiency.

2.1.2 The output distance function

Instead of looking at how much the input vector $x$ may be proportionally contracted with the output vector $y$ held fixed, the output distance function (ODF) introduced by Shephard (Reference Shephard1970), considers by how much the output vector may be proportionally expanded with the input vector held fixed. It is defined on the output set $P\left (x\right )$ or the production technology $T$ as follows:

\begin{equation*} D_{O}\left (x,y\right ) =\underset {\vartheta _{O}}{\min }\left \{ \vartheta _{O}:\frac {y}{\vartheta _{O}}\in P\left ( x\right ) \right \} =\underset {\vartheta _{O}}{\min }\left \{ \vartheta _{O}:\left ( x,\frac {y}{\vartheta _{O}}\right ) \in T\right \} \end{equation*}

where $1/\vartheta _{O}$ represents the proportional expansion of outputs required to reach the upper boundary of the output set or the production frontier, holding the inputs constant. The function $D_{O}\left ( x,y\right )$ is defined as the ratio of the observed output to maximum potential output obtainable from the given input. Therefore, for any output vector $y$ , the expression $y/D_{O}\left (x,y\right )$ represents the largest output vector on the ray from the origin through $y$ that can be produced by $x$ , as illustrated in Figure 2. If $y$ is on the boundary of the output set or the production frontier, then $D_{O}\left (x,y\right )=1$ , which implies that the producer operates at full technical efficiency. If $y$ is within the boundary of the output set or the production frontier, then $D_{O}\left (x,y\right )\lt 1$ , indicating that the producer operates with technical inefficiency.

The Debreu-Farrell output-oriented measure of technical efficiency is defined as follows:

\begin{equation*} TE_{O}\left ( x,y\right ) =\max _{\vartheta _{FO}}\left \{ \vartheta _{FO}:\vartheta _{FO}y\in P\left ( x\right ) \right \} =\max _{\vartheta _{FO} }\left \{ \vartheta _{FO}:\left ( x,\vartheta _{FO}y\right ) \in T\right \} \end{equation*}

The Debreu-Farrell output-oriented measure of technical efficiency is defined as the reciprocal of the ODF: $TE_{O}\left (x,y\right ) =\left [D_{O}\left (x,y\right )\right ]^{-1}$ . The expression $TE_{O}\left (x,y\right )\geq 1$ represents a radial expansion of outputs required to achieve efficiency; the greater this measure, the lower the efficiency. Technical inefficiency is defined as follows:

\begin{equation*} TI_{O}\left ( x,y\right ) =TE_{O}\left ( x,y\right ) -1=\frac {1}{D_{O}\left ( x,y\right ) }-1 \end{equation*}

where $TI_{O}\left ( x,y\right ) \geq 0.$ The output distance function has the following properties [see Färe and Grosskopf (Reference Färe, Grosskopf and Eichhorn1994) for more details]

1. i) representation, $D_{O}\left ( x,y\right ) \leq 1$ iff $y$ $\in P\left ( x\right )$ or $\left ( x,y\right ) \in T$

2. ii) non-increasing and quasi-convex in inputs, and

3. iii) non-decreasing, convex and linearly homogeneous in outputs, $D_{O}\left ( x,\lambda y\right ) =\lambda D_{O}\left ( x,y\right )$ , $\lambda \gt 0$ .

The output distance function has been applied across various sectors to accommodate multiple-output technologies and assess efficiency. In the banking sector, it has been used to evaluate the efficiency of banks in providing diverse services, such as loans and other financial services, and to identify banks that may not perform efficiently, as demonstrated by Cuesta and Orea (Reference Cuesta and Orea2002) and Almanidis et al. (Reference Almanidis, Karakaplan and Kutlu2019). In the public sector, the output distance function has been applied to measure the efficiency of various public services, such as education, and to determine how efficiently financial and human resources are used to deliver various educational activities, as discussed by Letti et al. (Reference Letti, Bittencourt and Vila2022). Additionally, it has been utilized to measure the efficiency of public hospitals in providing medical services. By focusing on the outputs, these functions assist in evaluating and improving healthcare delivery, as demonstrated by Devitt et al. (Reference Devitt, Zieba and Dineen2024). Furthermore, the ODF has been applied to address undesirable outputs. This is modeled by holding inputs $x$ constant and treating undesirable outputs, $b$ , as desirable outputs, $y$ ; $D_{O}\left ( x,y\right ) =\underset {\vartheta _{O}}{\min }\left \{ \vartheta _{O}:\left ( \frac {y}{\vartheta _{O}},\frac {b}{\vartheta _{O}}\right ) \in P\left ( x\right ) \right \} =\underset {\vartheta _{O}}{\min }\left \{ \vartheta _{O}:\left ( x,\frac {y}{\vartheta _{O}},\frac {b}{\vartheta _{O}}\right ) \in T\right \}$ . See, for example, Färe et al. (Reference Färe, Grosskopf, Lovell and Yaisawarng1993) and Färe et al. (Reference Färe, Grosskopf, Lovell and Pasurka1989). This approach credits producers for proportionally expanding both desirable and undesirable outputs to reach the production frontier. However, this is applicable only if the adjustability of both types of outputs is required. Producers have no control over reducing undesirable outputs without also reducing desirable outputs, and producing more desirable outputs requires producing more undesirable outputs, such as generating pollution as byproducts of producing desirable outputs. Furthermore, undesirable outputs are inevitably produced unless the entire production process is terminated.

Figure 2. The Debreu-Farrell output-oriented measure of technical efficiency.

The standard input and output distance functions adjust both desirable and undesirable outputs proportionally at the same rate. This approach may not align with the objectives of policy-makers who aim to simultaneously reduce undesirable outputs and increase desirable ones. Furthermore, these standard distance functions treat technical inefficiency as environmental inefficiency. Future studies that compare and differentiate these inefficiencies should contribute to a deeper understanding of their distinctions.

2.3.1 The directional technology distance function

The directional technology distance function (DTDF) generalizes Shephard’s input and output distance functions, providing a tool to address efficiency issues in an integrated manner. Chambers et al. (Reference Chambers, Chung and Färe1998) introduced it as a variant of the Luenberger (Reference Luenberger1995) shortage function. This function allows for the simultaneous expansion of $y$ and contraction of $x$ according to a specified direction vector $g$ $=\left (g_{x},g_{y}\right )$ , where $g_{x}\in R_{+}^{N}$ and $g_{y}\in R_{+}^{M}$ . This means that inputs are contracted in the direction of $g_{x}$ , and outputs are expanded in the direction of $g_{y}$ . In particular, the directional technology distance function is defined as follows:

(1) \begin{equation} \vec {D}_{T}\left ( x,y;\ g_{x},g_{y}\right ) =\max _{\theta _{T}}\left \{ \theta _{T}:\left ( x-\theta _{T}g_{x},y+\theta _{T}g_{y}\right ) \in T\right \} \end{equation}

Figure 4. The directional measure of technical inefficiency.

Efficient producers operating on the frontier of $T$ have $\vec {D}_{T}\left (x,y;\ g_{x},g_{y}\right )=0$ , indicating that no further expansion of outputs or contraction of inputs is feasible. Inefficiency is indicated by $\vec {D}_{T}\left (x,y;\ g_{x},g_{y}\right )\gt 0$ , with higher values denoting greater inefficiency when producers operate below the frontier of $T$ . Eliminating technical inefficiency for those operating at point A would move them to point $B=\left (x^{T},y^{T}\right )=\left (x-\theta _{T}g_{x},y+\theta _{T}g_{y}\right )$ on the frontier of $T$ , as illustrated in Figure 4. The DTDF serves as a technology-oriented measure of technical inefficiency.

\begin{equation*} TI_{T}=\vec {D}_{T}\left ( x,y;\ g_{x},g_{y}\right ) \end{equation*}

As noted by Chambers et al. (Reference Chambers, Chung and Färe1998), the directional technology distance function has the following properties:

1. i) representation, $\vec {D}_{T}\left ( x,y;\ g_{x},g_{y}\right ) \geq 0$ iff $\left (x,y\right ) \in T$

2. ii) translation, $\vec {D}_{T}\left ( x-\alpha g_{x},y+\alpha g_{y};\ g_{x},g_{y}\right ) =\vec {D}_{T}\left (x,y;\ g_{x},g_{y}\right ) -\alpha$ , for $\alpha \in R$

3. iii) non-decreasing in $x$ and non-increasing in $y$ if inputs and outputs are freely disposable

4. iv) concave in $\left (x,y\right )$

5. v) homogeneous of degree $-1$ in $g$ , That is, $\vec {D}_{T}\left (x,y;\ \lambda g_{x},\lambda g_{y}\right )=\lambda ^{-1}\vec {D}_{T}\left ( x,y;\ g_{x},g_{y}\right )$ , for $\lambda \gt 0$ , and

6. vi) homogeneous of degree $+1$ in $x$ and $y$ if the technology exhibits constant returns to scale, $\vec {D}_{T}\left (\lambda x,\lambda y;\ g_{x},g_{y}\right ) =\lambda \vec {D}_{T}\left ( x,y;\ g_{x},g_{y}\right )$ , for $\lambda \gt 0$ .

The directional technology distance function can be utilized to represent technologies that are not separable in desirable and undesirable outputs. This is useful in these production contexts because it allows for non-radial or hyperbolic expansions of desirable outputs and contractions of undesirable outputs, as demonstrated by Malikov et al. (Reference Malikov, Kumbhakar and Tsionas2016). The DTDF has been applied across various sectors to assess efficiency. In the banking sector, these functions have been utilized to determine bank inputs and outputs and to evaluate the performance and efficiency of banks by assessing how efficiently they convert inputs-such as capital, labor, and other resources-into outputs, like loans and other financial services, while minimizing undesirable outputs, such as non-performing loans. For examples, see Koutsomanoli-Filippaki et al. (Reference Koutsomanoli-Filippaki, Margaritis and Staikouras2012), Guarda et al. (Reference Guarda, Rouabah and Vardanyan2013), and Huang and Chung (Reference Huang and Chung2017). In the public sector, the DTDF has been employed to measure the efficiency of public hospitals in delivering multiple health services by utilizing their resources, such as medical staff, equipment, and other resources, and minimizing undesirable outcomes. For examples, see Vardanyan et al. (Reference Vardanyan, Valdmanis, Leleu and Ferrier2022).

2.3.2 The directional input distance function

The inefficiency measures derived from the directional distance function depend on the directional vector, $g=\left (g_{x},g_{y}\right )$ . By setting $g_{y}=0$ , the directional vector becomes $g$ $=\left (g_{x},0\right )$ , which allows for input contraction with outputs held constant, as illustrated in Figure 4. In this context, equation (1) transforms into the directional input distance function (DIDF), permitting only input contraction; $\vec {D}_{T}\left (x,y;\ g_{x},0\right )=\vec {D}_{I}\left (y,x;\ g_{x}\right )$

\begin{equation*} \vec {D}_{I}\left ( y,x;\ g_{x}\right ) =\max _{\theta _{I}}\left \{ \theta _{I}:\left ( x-\theta _{I}g_{x}\right ) \in L\left ( y\right ) \right \} =\max _{\theta _{I}}\left \{ \theta _{I}:\left ( x-\theta _{I}g_{x}\text {, }y\right ) \in T\right \} \end{equation*}

Moreover, according to Chambers et al. (Reference Chambers, Chung and Färe1996, Reference Chambers, Chung and Färe1998) and Färe and Grosskopf (Reference Färe and Grosskopf2000), if the directional input vector, $g_{x}$ , equals the observed input vector, $x$ , $\left ( \text {that is, }g_{x}=-x\right )$ , then

\begin{equation*} \vec {D}_{I}\left (y,x;\ g_{x}\right )=\vec {D}_{I}\left (y,x;\ -x\right ) =1-\frac {1}{D_{I}\left (y,x\right )} \end{equation*}

In this context, a relationship exists between the DIDF, $\vec {D}_{I}\left (y,x;\ -x\right )$ , and the Shephard input distance function, $D_{I}\left (y,x\right )$ . As illustrated in Figure 4, producers operating at point $A$ can maintain constant output while reducing input in the direction of $g_{x}=-x$ to reach point $I$ . The directional input distance function serves as an input-oriented measure of technical inefficiency.

\begin{equation*} TI_{I}=\vec {D}_{I}\left ( y,x;\ g_{x}\right ) \end{equation*}

The directional input distance function satisfies the following properties, as outlined by Chambers et al. (Reference Chambers, Chung and Färe1996):

1. i) representation, $\vec {D}_{I}\left ( y,x;\ g_{x}\right ) \geq 0$ iff $x$ $\in L\left ( y\right )$ or $\left ( x,y\right ) \in T$

2. ii) translation, $\vec {D}_{I}\left ( y,x-\alpha g_{x};\ g_{x}\right ) =\vec {D}_{I}\left ( y,x;\ g_{x}\right ) -\alpha$ , for $\alpha \in R$

3. iii) concavity in inputs

4. iv) positive monotonicity in inputs. That is, $x^{\prime }\gt$ $x$ implies $\vec {D}_{I}\left ( y,x^{\prime };\ g_{x}\right ) \geq \vec {D}_{I}\left ( y,x;\ g_{x}\right )$

5. v) negative monotonicity in outputs. That is, $y^{\prime }\gt$ $y$ implies $\vec {D}_{I}\left ( y^{\prime },x;\ g_{x}\right ) \leq \vec {D}_{I}\left ( g,x;\ g_{x}\right )$ , and

6. vi) homogeneity of degree $-1$ in $g_{x}$ . That is, $\vec {D}_{I}\left ( y,x;\ \lambda g_{x}\right ) =\lambda ^{-1}\vec {D}_{I}\left ( y,x;\ g_{x}\right )$ , for $\lambda \gt 0$ .

The directional input distance function has been applied across various sectors to evaluate efficiency. In the public sector, researchers have utilized it to compare the performance of public hospitals with that of hospitals in other countries. This comparison assesses how efficiently hospitals utilize their inputs compared to their peers, identifying best practices and areas for improvement, as demonstrated by Dervaux et al. (Reference Dervaux, Ferrier, Leleu and Valdmanis2004). Furthermore, the DIDF has been employed to evaluate the efficiency of dairy farming by analyzing the use of various resources, such as labor, equipment, and other inputs, to determine how efficiently these resources are utilized to produce dairy products, as shown by Serra et al. (Reference Serra, Lansink and Stefanou2011).

2.3.3 The directional output distance function

By setting $g_{x}=0$ , the directional vector becomes $g$ $=\left ( 0,g_{y}\right )$ , which allows for output expansion while inputs remain constant, as illustrated in Figure 4. In this context, equation (1) simplifies to the directional output distance function (DODF), allowing solely for the expansion of output; $\vec {D}_{T}\left (x,y;\ 0,g_{y}\right )=\vec {D}_{O}\left (x,y;\ g_{y}\right )$

\begin{equation*} \vec {D}_{O}\left ( x,y;\ g_{y}\right ) =\max _{\theta _{o}}\left \{ \theta _{O}:\left ( y+\theta _{O}g_{y}\right ) \in P\left ( x\right ) \right \} =\max _{\theta _{o}}\left \{ \theta _{O}:\left ( x,y+\theta _{O}g_{y}\right ) \in T\right \} \end{equation*}

Moreover, as noted by Chambers et al. (Reference Chambers, Chung and Färe1998) and Färe and Grosskopf (Reference Färe and Grosskopf2000), if the directional output vector, $g_{y}$ , equals the observed output vector, $y$ $\left (\text {that is, }g_{y}=y\right )$ , then

\begin{equation*} \vec {D}_{O}\left ( x,y;\ g_{y}\right ) =\vec {D}_{O}\left ( x,y;\ y\right ) =\frac {1}{D_{O}\left ( x,y\right ) }-1 \end{equation*}

In this context, a relationship exists between the DODF, $\vec {D}_{O}\left ( x,y;\ y\right )$ , and the Shephard output distance function, $D_{O}\left ( x,y\right )$ . As illustrated in Figure 4, producers operating at point $A$ can maintain constant input while expanding output in the direction of $g_{y}=$ $y$ to reach point $O$ . The directional output distance function serves as an output-oriented measure of technical inefficiency.

\begin{equation*} TI_{O}=\vec {D}_{O}\left ( x,y;\ g_{y}\right ) \end{equation*}

The directional output distance function satisfies the following properties, as outlined by Färe et al. (Reference Färe, Grosskopf, Noh and Weber2005):

1. i) representation, $\vec {D}_{O}\left ( x,y;\ g_{y}\right ) \geq 0$ iff $y$ $\in P\left ( x\right )$ or $\left ( x,y\right ) \in T$

2. ii) translation, $\vec {D}_{O}\left ( x,y+\alpha g_{y};\ g_{y}\right ) =\vec {D}_{O}\left ( x,y;\ g_{y}\right ) -\alpha$ , for $\alpha \in R$

3. iii) concavity in outputs

4. iv) negative monotonicity in outputs. That is, $y^{\prime }\gt$ $y$ implies $\vec {D}_{O}\left ( x,y^{\prime };\ g_{y}\right ) \leq \vec {D}_{O}\left ( x,y;\ g_{y}\right )$ , and

5. v) homogeneity of degree minus $-1$ in $g_{y}$ . That is, $\vec {D} _{O}\left ( x,y;\ \lambda g_{y}\right ) =\lambda ^{-1}\vec {D}_{O}\left ( x,y;\ g_{y}\right )$ , for $\lambda \gt 0$ .

The directional output distance function has been applied across various agricultural and industrial sectors to accommodate multiple-output technologies, assess efficiency, and mitigate undesirable outputs such as pollution and emissions. This function is employed to address these undesirable outputs, $b$ , with the DODF defined as follows: $\vec {D}_{O}\left ( x,y,b;\ g_{y}, g_{b} \right ) =\max _{\theta _{o}}\left \{ \theta _{O}:\left ( y+\theta _{O}g_{y}, b-\theta _{O}g_{b}\right ) \in P\left ( x\right ) \right \} = \max _{\theta _{o}}\left \{ \theta _{O}:\left ( x,y+\theta _{O}g_{y}, b-\theta _{O}g_{b}\right ) \in T\right \}$ ; see, for example, Färe et al. (Reference Färe, Grosskopf and Weber2006), Watanabe and Tanaka (Reference Watanabe and Tanaka2007), Feng et al. (Reference Feng, Wang and Serletis2018), and Yang et al. (Reference Yang, Cheng, Zou and Geng2021).

2.3.4 The directional vector

The measure of inefficiency derived from directional distance functions depends on the choice of the direction vector $g$ , which projects the data onto the frontier of $T$ . Inefficiency is measured by selecting either an exogenous or an endogenous direction vector. The exogenous vector is pre-specified, whereas the endogenous vector determines the direction based on specific internal behavior.

2.3.4.2 Endogenous directional vector

An endogenous direction vector is selected to guide any producer towards the benchmark of minimizing costs or maximizing revenue/profit. This choice depends on the availability of price information and the fulfillment of necessary behavioral assumptions, specifically the behaviors of minimizing costs or maximizing revenue/profit.

Using the direction vector $g$ $=\left (g_{x_{1}}^{C},g_{x_{2}}^{C}\right )$ , the directional measure of input inefficiency is determined by projecting any inefficient producer onto the cost-minimizing bundle $ C$ , where producers achieve both technical and allocative efficiency. Allocative inefficiency arises from the failure to choose the cost-minimizing input vector given the relative input market prices. For an endogenous direction vector that projects to the cost-minimizing benchmark, see, for example, Malikov et al. (Reference Malikov, Kumbhakar and Tsionas2016). Using the direction vector $g$ $=\left ( g_{y_{1}}^{R},g_{y_{2}}^{R}\right )$ , the directional measure of output inefficiency is determined by projecting any inefficient producer onto the revenue-maximizing bundle $ R$ , where producers achieve both technical and allocative efficiency. Allocative inefficiency arises from the failure to choose the revenue-maximizing output vector given the relative output market prices. Esheba (Reference Esheba2018) and Esheba and Serletis (Reference Esheba and Serletis2023) provided a set of directions consistent with cost minimization, as well as revenue and profit maximization. Using the direction vector $g$ $=\left (g_{x}^{\pi },g_{y}^{\pi }\right )$ , the directional measure of overall inefficiency is determined by projecting any inefficient producer to the profit-maximizing bundle $\pi$ , where producers achieve both technical and allocative efficiency. Allocative inefficiency arises from the failure to choose the profit-maximizing input-output vector given the relative input and output market prices. All profit inefficiency for producers operating below the profit frontier can be regarded as measures of overall technical inefficiency, as noted by Zofio et al. (Reference Zofio, Pastor and Aparicio2013). Feng et al. (Reference Feng, Wang and Serletis2018) discussed an endogenous direction vector projecting to the profit-maximizing benchmark, and Atkinson and Tsionas (Reference Atkinson and Tsionas2016) provided a set of directions consistent with cost minimization and profit maximization.

Färe et al. (Reference Färe, Grosskopf and Whittaker2013) developed a method for selecting direction vectors that are endogenously determined based on exogenous normalization constraints using Data Envelopment Analysis, without the need for price data. Hampf and Kruger (Reference Hampf and Kruger2015) utilized optimization methods to endogenously determined optimal directions for a non-parametric efficiency analysis. However, further research is necessary to compare alternative choices of directional vectors and to establish a framework for determining an optimal set of directions using parametric measurement methods.

Figure 5. Radial, hyperbolic and directional measures of technical inefficiency.

To summarize, Figure 5 illustrates the projections of the observed input-output vector at point $ A$ using various types of distance functions. The standard distance function projects $ A$ proportionally onto $ R$ . The radial measure of efficiency is given by $TE_{R}\left (x,y\right )=\max _{\vartheta }\left \{\vartheta :\left (\vartheta x,\vartheta y\right )\in T\right \}$ . However, the radial measure provided by the standard distance function can yield high inefficiency measures even when the observed input-output vector is very close to the frontier, as noted by Hudgins and Primont (Reference Hudgins, Primont, Hudgins and Primont2007). The hyperbolic distance function projects $A$ hyperbolically onto $H$ , where the intersection between the hyperbolic curve and the frontier of $T$ is the point $H=\left (\vartheta _{FH}x,y/\vartheta _{FH}\right )$ . The hyperbolic measure of efficiency is expressed as $TE_{H}\left (x,y\right )=\max _{\vartheta _{FH}}\left \{\vartheta _{FH}:\left (x/\vartheta _{FH},\vartheta _{FH}y\right )\in T\right \}$ . Implementing the hyperbolic measure using the hyperbolic distance function can be complex due to the non-linear optimization involved. The directional technology distance function is particularly well-suited for simultaneously contracting $ x$ and expanding $ y$ to project $ A$ onto $ D$ using the direction vector $ g$ . The directional measure of inefficiency is technology-oriented and expressed as $\vec {D}_{T}\left (x,y;\ g_{x},g_{y}\right )=\max _{\theta _{T}}\left \{ \theta _{T}:\left (x-\theta _{T}g_{x},y+\theta _{T}g_{y}\right )\in T\right \}$ . The important properties of the alternative distance functions that can be used for measuring inefficiency and the relationships among them are summarized in Table 1.

Table 1. A summary of the important properties of alternative distance functions

3.1.1 The cross-sectional models

The early literature on stochastic frontier models utilized cross-sectional models, where specific distributions for inefficiency and error terms were assumed to estimate the frontier function. These distribution assumptions were also necessary to separate inefficiency from the error term.

The stochastic production frontier was independently proposed by Aigner et al. (Reference Aigner, Lovell and Schmidt1977) and Meeusen and Van den Broeck (Reference Meeusen and Van den Broeck1977). Battese and Cora (Reference Battese and Cora1977) introduced the first application. It is presented as follows:

\begin{equation*} Y_{i}=\alpha +f(X_{i};\ \beta )+\nu _{i}-u_{i} \end{equation*}

In this context, inputs, outputs, stochastic factors, and inefficiency vary only across producers. A key issue with the cross-sectional model is its reliance on the strong assumption that inefficiencies are independent of the regressors. If this assumption is violated, it results in inconsistent estimates of the model’s parameters and the measures of inefficiency.

3.1.2 The panel data models

The use of panel data addresses the limitations of cross-sectional models and offers several advantages in models with time-invariant inefficiency. It provides consistent estimates of inefficiency by incorporating more temporal observations for the same producer, assuming the time series is sufficiently large. Additionally, it does not require inefficiency to be independent of the regressors, which is beneficial for including time-invariant regressors in the model. Furthermore, there is no need to make specific distributional assumption regarding inefficiency, and all parameters of the model can be estimated using standard estimation procedures for panel data, such as fixed and random-effects. The production frontier panel data model can be represented as follows:

\begin{equation*} Y_{it}=\alpha +f(X_{it};\ \beta )+\nu _{it}-u_{i} \end{equation*}

In this context, inputs, outputs, and stochastic factors vary over time and among producers, but inefficiency varies only among producers, this assumption may be unrealistic. However, it might hold true if the time dimension of the panel is particularly short or if inefficiency is associated with management and there is no change in management during that period. If the time dimension is long, it seems unrealistic to assume constant inefficiency over time or for inefficient producers to continue operating in the market.

3.1.2.1 The fixed-effects models

If inefficiency is considered systematic and, therefore, $ u_{i}$ is treated as a producer-specific constant or an unknown fixed parameter to be estimated, a fixed-effect model can be implemented. No distribution assumption is required for $ u_{i}$ , which is assumed to be correlated with the regressors $ X_{it}$ or the random errors $ \nu _{it}$ . Schmidt and Sickles (Reference Schmidt and Sickles1984) defined a fixed-effects model of inefficiency as follows:

\begin{equation*} Y_{it}=\alpha _{i}+f(X_{it};\ \beta )+\nu _{it} \end{equation*}

Since $ u_{i}$ is treated as fixed, it becomes the producer-specific intercept $ \alpha _{i} = \alpha - u_{i}$ . These are regarded as fixed numbers that can be estimated as parameters or eliminated through suitable transformation if the number of producers is too large. Schmidt and Sickles (Reference Schmidt and Sickles1984) considered various procedures for estimating the fixed-effects model: the within estimator, the Generalized Least Squares (GLS) estimator, and the Maximum Likelihood Estimation (MLE). The within estimator does not assume independence between $ u_{i}$ and the regressors, whereas the GLS estimator assumes that $ u_{i}$ is uncorrelated with the regressors. The MLE assumes both distributional and independence assumptions. Sickles (Reference Sickles2005) presented a diverse array of methods for identifying producer-specific inefficiency using panel estimators. Koop et al. (Reference Koop, Osiewalski and Steel1997) described procedures for Bayesian estimation of fixed-effects inefficiency models.

Using the within estimator, the fixed-effects estimate $\widehat {\beta }$ , also known as the within estimate, can be determined by either regressing $\widetilde {Y}_{it}=\left ( y_{it}-\overline {y}_{i}\right )$ on $\widetilde {X}_{it}=\left ( x_{it}-\overline {x}_{i}\right )$ , where $\overline {y}_{i}= {\textstyle \sum \nolimits _{i=1}^{T}} y_{it}/T$ and $\overline {x}_{i}= {\textstyle \sum \nolimits _{i=1}^{T}} x_{it}/T$ , thereby eliminating $\alpha _{i}$ , or, equivalently, by regressing $Y_{it}$ on $X_{it}$ along with a set of specific dummy variables for producers using the Ordinary Least Squares (OLS). Consequently, $\widehat {\alpha }_{i}$ is obtained by averaging its residuals over time as $\widehat {\alpha }_{i}=\overline {y}_{i}-\overline {x}_{i}\widehat {\beta }$ , or, equivalently, $\widehat {\alpha }_{i}$ represents the estimated coefficients of the dummy variables. Inefficiency is assessed by comparing the estimated intercept of each producer to the maximum estimated value.

\begin{equation*} \widehat {u}_{i}=\max \nolimits _{j}\left \{\widehat {\alpha }_{j}\right \}-\widehat {\alpha }_{i} \end{equation*}

Producer-specific efficiency can be obtained from $TE_{i}=\exp \left (-\widehat {u}_{i}\right )$ . However, this approach considers the producer with the highest intercept as fully efficient, making inefficiency for other producers relative to that producer. Feng and Horrace (Reference Feng and Horrace2012) estimated inefficiency relative to the least efficient producer instead of the most efficient one by comparing the estimated intercept of each producer to the minimum estimated value. They argued that these inefficiency estimates have smaller bias than those using the maximum estimated value when many producers operate close to the efficient frontier. However, in both cases, inefficiency is estimated as relative rather than absolute inefficiency. Furthermore, the intercept $\widehat {\alpha }_{i}$ captures all time-invariant unobserved heterogeneity, not only those related to inefficiency. Additionally, as pointed out by Kim and Schmidt (Reference Kim and Schmidt2000), Wang and Schmidt (Reference Wang and Schmidt2009), and Satchachai and Schmidt (Reference Satchachai and Schmidt2010), the estimation of inefficiency based on the fixed-effects estimator can be upwardly biased when the number of time series is small and the number of cross-sectional observations is large. The max operator induces an upward bias in $\widehat {\alpha }=\max \nolimits _{j}\left \{\widehat {\alpha }_{j}\right \}$ , which in turn induces an upward bias in the inefficiency estimates $\widehat {u}_{i}$ .

Wikstrom (Reference Wikstrom2016) introduced a modified fixed-effects estimator that does not suffer from bias in large cross-sectional observations. This is accomplished by utilizing the second central moment of the inefficiency distribution to correct the intercept value derived from the fixed-effects estimator. Wikstrom presented a consistent estimator of $\alpha$ , assuming a half-normal distribution and an exponential distribution for $u_{i}$ as $\widehat {\alpha }=\widehat {\mu }_{\alpha }+\widehat {\mu }_{u}$ where $\widehat {\mu }_{\alpha }= {\textstyle \sum \nolimits _{i=1}^{N}} \widehat {\alpha }_{i}/N$ , $\widehat {\mu }_{u}=\widehat {\sigma }_{\alpha } ^{2}\left ( 2/\pi -2\right ) ^{1/2}$ assuming a half-normal distribution for $u_{i}$ and $\widehat {\mu }_{u}=\left ( \widehat {\sigma }_{\alpha }^{2}\right ) ^{1/2}$ assuming an exponential distribution for $u_{i}$ , $\widehat {\sigma }_{\alpha }^{2}=\left ( {\textstyle \sum \nolimits _{i=1}^{N}} \left ( \widehat {\alpha }_{i}-\widehat {\mu }_{\alpha }\right ) ^{2}/N\right ) -\left ( \widehat {\sigma }_{v}^{2}/T\right )$ . The modified fixed-effects estimator of $u_{i}$ is defined as follows:

\begin{equation*} \widehat {u}_{i}=\widehat {\alpha }-\widehat {\alpha }_{i} \end{equation*}

The fixed-effects estimator has the advantage of not requiring a distributional assumption on inefficiency and allows inefficiency to be correlated with any other variables. However, the assumption of time invariance in inefficiency is very restrictive and may not be reasonable for relatively long panels. Additionally, fixed-effect time-invariant models are based on the assumption that all time-invariant effects are parts of inefficiency. Consequently, inefficiency measures include any other source of time-invariant unobserved heterogeneity, not only those related to inefficiency, making it challenging to distinguish unobserved heterogeneity from inefficiency, as noted by Greene (Reference Greene2004a). Furthermore, time-invariant regressors cannot be used in the specification of the frontier model, as this would lead to perfect multicollinearity between $ \alpha _{i}$ and the time-invariant regressors.

3.2.1 The fixed-effects models

Cornwell et al. (Reference Cornwell, Schmidt and Sickles1990) modified the assumption of time invariance in the Schmidt and Sickles (Reference Schmidt and Sickles1984) model by replacing $\alpha _{i}$ with a quadratic function of time. This adjustment allows inefficiency to vary over time and among individual producers. The model is represented as follows:

\begin{equation*} Y_{it}=\alpha _{it}+f(X_{it};\ \beta )+v_{it} \end{equation*}

where $\alpha _{it}=\theta _{0i}+\theta _{1i}t+\theta _{2i}t^{2}$ . The model is estimated by regressing the residuals for each producer $\left ( \widehat {\varepsilon }_{it}=Y_{it}-X_{it}^{^{\prime }}\widehat {\beta }\right )$ on a constant, time, and time squared. The fitted values from this regression provide an estimate of $ \alpha _{it}$ . Inefficiency measures are computed relative to the most efficient producer across all time periods or, alternatively, to the most efficient producer in a given year. This modification allows the most efficient producer to change from year to year.

\begin{equation*} \widehat {u}_{it}=\max \nolimits _{j}\left \{ \widehat {\alpha }_{jt}\right \} -\widehat {\alpha }_{it} \end{equation*}

The advantages of this model include its independence of distribution assumption on inefficiency and its allowance of inefficiency to vary among producers and over time. However, it is quite restrictive in describing the temporal behavior of inefficiency, as this is assumed to be deterministic. Additionally, this model cannot distinguish inefficiency from technical change, which represents a shift in the production frontier, because time is a factor in the inefficiency function.

Lee and Schmidt (Reference Lee, Schmidt, Lee and Schmidt1993) described inefficiency as the product of individual producer inefficiency and time effects; $\alpha _{it}=\theta _{t}\alpha _{i}$ , where $\theta _{t}= {\textstyle \sum \nolimits _{t}} \delta _{t}$ with $\delta _{t}$ being a dummy variable for each period $t$ and $\widehat {u}_{it}=\max \nolimits _{j}\left \{ \widehat {\theta }_{t} \widehat {\alpha }_{i}\right \} -\widehat {\theta }_{t}\widehat {\alpha }_{i}$ . This specification differs from the time-invariant fixed-effect model by allowing inefficiency to vary over time. However, the producer effect, denoted as $ \alpha _{i}$ , cannot be identified separately from $ \theta$ unless a specification for the inefficiency component is considered without the intercept. Furthermore, the temporal behavior of inefficiency is assumed to be the same for all producers.

The primary challenge with these fixed-effects, time-variant inefficiency models is that they require the estimation of numerous parameters, which can be constrained by very short panels. Furthermore, inefficiency varies over time in both models through the use of a time trend or time dummies, which hinders the control of technical change.

Greene (Reference Greene2005a, Reference Greene2005b) introduced what is known as the true fixed-effects model. It is represented as follows:

\begin{equation*} Y_{it}=\alpha _{i}+f(X_{it};\ \beta )+v_{it}-u_{it} \end{equation*}

In this context, $ \alpha _{i}$ represents unobserved time-invariant heterogeneity and is treated as a random variable that is correlated with $ X_{it}$ , but does not capture inefficiency and can be estimated as a parameter. A true fixed-effects model can be estimated by adding dummy variables for each producer to the model. The disadvantage of this model is that it induces the incidental parameters problem, which can lead to inconsistency because the number of parameters to be estimated depends on the number of producers, as discussed by Neyman and Scott (Reference Neyman and Scott1948). Recent studies have considered addressing the problem of incidental parameters in the true fixed-effects model by using within transformation to eliminate the producer effects for unobserved heterogeneity,Footnote ¹ see, for example, Wang and Ho (Reference Wang and Ho2010) and Chen, et al. (Reference Chen, Schmidt and Wang2014).

3.2.2 The random-effects models

In these models, time-variant inefficiency can either be independently and identically distributed (iid) among producers and over time, or it can be modeled as the product of a deterministic function of time, $ g(t;\ \gamma )$ , and a non-negative time-invariant random variable $ u_{i}$ , such that $ u_{it} = g(t;\ \gamma )u_{i}$ , where $ \gamma$ is a parameter to be estimated. Thus, $ g(t;\ \gamma )$ allows the data to determine the temporal behavior of inefficiency rather than imposing it a priori. Inefficiency is then estimated from $ \widehat {u}_{it} = \widehat {g}(t;\ \gamma )E(u_{i} \mid \varepsilon _{i})$ , or, alternatively, $ \widehat {u}_{i} = E(u_{i} \mid \varepsilon _{it})$ ; see Kumbhakar and Lovell (Reference Kumbhakar and Lovell2000).

Kumbhakar (Reference Kumbhakar1990) assumed that $u_{it}=u_{i}\left (1+\exp \left ( \gamma _{1}t+\gamma _{2}t^{2}\right )\right )^{-1}$ . His model allows inefficiency to either increase or decrease monotonically, depending on the values of $\gamma _{1}$ and $\gamma _{2}$ . Battese and Coelli (Reference Battese and Coelli1992) and Battese and Tessema (Reference Battese and Tessema1993) assumed that $u_{it}=\exp \left (-\gamma \left (t-T\right ) \right ) u_{i}$ , where $T$ represents the final time period. Their model implies that inefficiency changes over time in a monotonic manner, increasing or decreasing exponentially for all producers based on the sign of $ \gamma$ . Specifically, inefficiency increases at a decreasing rate when $ \gamma$ is positive and decreases at an increasing rate when $ \gamma$ is negative. A time-invariant model is obtained when $\gamma$ is equal to zero. Kumbhakar and Wang (Reference Kumbhakar and Wang2005) assumed that $u_{it}=\exp \left (-\gamma \left ( t-\underline {t}\right ) \right ) u_{i}$ . In this context, inefficiency evolves over time according to $\exp \left (-\gamma \left (t-\underline {t}\right )\right )$ , where $\underline {t}$ denotes the initial time period, ensuring $u_{it}=u_{i}$ at time $\underline {t}$ . A promising area for further research is developing a model that incorporates both the initial and final time periods, as it considers both market entry and exit in defining the reference point.

Lee and Schmidt (Reference Lee, Schmidt, Lee and Schmidt1993) assumed that $u_{it}=\gamma _{t}u_{i}$ , where $\gamma _{t}$ represents the parameters associated with the time dummy variables that need to be estimated. While Battese and Coelli (Reference Battese and Coelli1992) and Battese and Tessema (Reference Battese and Tessema1993) assumed a quite restrictive temporal behavior of inefficiency which is assumed to be the same for all producers, Cuesta (Reference Cuesta2000) modified the Battese and Coelli (Reference Battese and Coelli1992) model to allow for greater flexibility in how inefficiency changes over time by assuming $u_{it}=\exp \left (-\gamma _{i}\left ( t-T\right )\right )u_{i}$ , $u_{it}=\exp \left (g_{i}\left (t,T,z_{it}\right )\right )u_{i}$ . This specification allows inefficiency to evolve over time at varying rates among producers.

Cuesta and Orea (Reference Cuesta and Orea2002) and Feng and Serletis (Reference Feng and Serletis2009) extended the Battese and Coelli (Reference Battese and Coelli1992) model by assuming $u_{it}=\exp \left (-\gamma _{1}\left (t-T\right )-\gamma _{2}\left (t-T\right )^{2}\right )u_{i}$ . This specification relaxes the monotonicity of the time path of inefficiency using a two-parameter approach. Consequently, the model allows for producer effects to be either convex or concave and to vary over time, increasing in some periods and decreasing in others.

The primary advantage of random-effects models over fixed-effects time-variant inefficiency models is that they allow for the inclusion of time-invariant regressors. However, random-effects models require independence between inefficiency and regressors in the model, a condition not required in fixed-effects models.

The Hausman and Taylor (Reference Hausman and Taylor1981) model integrates fixed and random-effects models, allowing inefficiency to be uncorrelated with certain, but not all, regressors. It also enables the inclusion of time-invariant regressors in the model. In this framework, producer inefficiency can be consistently estimated and separated from the producer effects or the intercept, provided that the cross-sectional and temporal observations are sufficiently large.

To separate producer heterogeneity, or producer effects, and inefficiency, where inefficiency can vary over time and may be either iid or a function of exogenous variables, Greene (Reference Greene2005a, Reference Greene2005b) introduced a time-invariant random effect to account for unobserved producer heterogeneity and proposed what is known as the true random-effects model.

\begin{equation*} Y_{it}=\left ( \alpha +w_{i}\right ) +f(X_{it};\ \beta )+v_{it}-u_{it} \end{equation*}

where $\alpha _{i}=\alpha +w_{i}$ represents unobserved time-invariant heterogeneity and is treated as a random variable that is uncorrelated with $X_{it}$ . It is important to note that Kumbhakar and Wang (Reference Kumbhakar and Wang2005) introduced these producer-specific intercepts $\alpha _{i}$ to account for heterogeneous technologies. If $\alpha _{i}$ is treated as a random variable that is correlated with $X_{it}$ but does not capture inefficiency, then the model becomes the true fixed-effects model.

3.8.1 Determinants of inefficiency models

In these models, inefficiency is represented as a function of explanatory variables $ Z$ that reflect characteristics specific to each producer and explain the differences in inefficiency among them. For further discussion, see Deprins and Simar (Reference Deprins and Simar1989), Kumbhakar et al. (Reference Kumbhakar, Ghosh and McGuckin1991), and Huang and Liu (Reference Huang and Liu1994) who introduced interaction terms between producer-specific characteristics and regressors, $ z_{i}x_{i}$ .

\begin{equation*} u_{i}=g\left ( z_{i},z_{i}x_{i};\ \delta \right ) +e_{i}\end{equation*}

where $\delta$ represents unknown parameters to be estimated, and $e_{i}$ is a random variable defined by the truncation of a normal distribution. If there are no interaction terms $z_{i}x_{i}$ , the model reduces to those of Deprins and Simar (Reference Deprins and Simar1989) and Kumbhakar et al. (Reference Kumbhakar, Ghosh and McGuckin1991). Tsionas (Reference Tsionas2006) extended the Kumbhakar et al. (Reference Kumbhakar, Ghosh and McGuckin1991) model to a panel data model that allows for dynamic inefficiency; $\ln u_{it}=z_{it}\delta +\rho \ln u_{i,t-1}+e_{it}$ . Srairi (Reference Srairi2010) further extended the Kumbhakar et al. (Reference Kumbhakar, Ghosh and McGuckin1991) model to panel data where $u_{it}=g\left ( z_{it};\ \delta \right ) +e_{it}$ to examine bank-specific variables that may explain the sources of inefficiency.

Determinants of inefficiency models face the challenge of ensuring non-negative inefficiency values. Kumbhakar et al. (Reference Kumbhakar, Ghosh and McGuckin1991) proposed a solution to address this issue: $u_{i}=\left \vert N\left (z_{i}\delta, \sigma _{u}^{2}\right )\right \vert$ . Reifschneider and Stevenson (Reference Reifschneider and Stevenson1991) assumed that $u_{i}=u_{i}^{\ast }+\exp \left ( z_{i}\delta \right )$ , where both $u_{i}^{\ast }\thicksim N^{+}(0,\sigma _{u}^{2})$ and $\exp \left ( z_{i}\delta \right )$ are positive. However, it is not necessary for both components to be positive to achieve a positive $ u_{i}$ . Huang and Liu (Reference Huang and Liu1994) extended the assumption of Reifschneider and Stevenson (Reference Reifschneider and Stevenson1991) by assuming only that $u_{i}^{\ast }\geq -\exp \left (z_{i}\delta \right )$ .

3.8.2 Determinants of inefficiency distribution models

In these models, producer-specific characteristics can be incorporated into the mean, variance, or both parameters of the inefficiency distribution. Battese and Coelli (Reference Battese and Coelli1995) and Wang and Ho (Reference Wang and Ho2010) proposed that the mean of the inefficiency distribution be modeled as a function of explanatory variables that represent producer-specific characteristics.

\begin{equation*} u_{it}=g\left ( z_{it};\ \delta \right ) u_{i}\text {, }u_{it}\thicksim N^{+}(\mu _{it},\sigma _{u}^{2})\text {, }\mu _{it}=z_{it}\delta _{m}\end{equation*}

Including producer-specific characteristics in the variance of the inefficiency distribution is motivated by the potential presence of heteroscedasticity in inefficiency. Reifschneider and Stevenson (Reference Reifschneider and Stevenson1991), Caudill and Ford (Reference Caudill and Ford1993), and Caudill et al. (Reference Caudill, Ford and Gropper1995) assumed that the inefficiency term, $u$ , is heteroskedastic and included the standard deviation in exponential form to ensure a positive estimate of the variance parameter for all parameters involved, $Z$ and $\gamma _{u}$ .

\begin{equation*} \sigma _{ui}=\exp ( z_{ui}\gamma _{u} )\text {, }\sigma _{vi}=\exp ( \gamma _{v} ) \end{equation*}

It is also possible to assume that both $ u$ and $ v$ are heteroskedastic, which is referred to in the literature as the doubly heteroskedastic model. The variance parameters of the $ u$ and $ v$ distributions are modeled as functions of the explanatory variables $ z_{ui}$ and $ z_{vi}$ , which may or may not be equivalent; see, for example, Hadri (Reference Hadri1999) and Hadri et al. (Reference Hadri, Guermat and Whittaker2003).

\begin{equation*} \sigma _{ui}=\exp \left ( z_{ui}\gamma _{u}\right ) \text {, }\sigma _{vi}=\exp \left ( z_{vi}\gamma _{v}\right ) \end{equation*}

Including producer-specific characteristics in both the mean and the variance of the inefficiency distribution allows for non-monotonic inefficiency effects across producers. For examples, see Wang (Reference Wang2002) and Wang and Schmidt (Reference Wang and Schmidt2002).

\begin{equation*} u_{it}\thicksim N^{+}(\mu _{it},\sigma _{uit}^{2})\text {, }\mu _{it}=z_{it}\delta _{m}\text {, }\sigma _{uit}^{2}=\exp \left ( z_{it}\gamma _{u}\right ) \end{equation*}

Kumbhakar and Wang (Reference Kumbhakar and Wang2005) proposed that the variance parameter of the $ v$ distribution could be modeled as a function of the explanatory variables $ z_{vi}$ , alongside the mean and variance of the inefficiency distribution.

\begin{equation*} u_{i}\thicksim N^{+}(\mu _{i},\sigma _{ui}^{2})\text {, }\mu _{i}=z_{i}\delta _{m}\text {, }\sigma _{ui}^{2}=\exp \left ( z_{ui}\gamma _{u}\right ) \text {, }v_{it}\thicksim N(0,\sigma _{vi}^{2})\text {, }\sigma _{vi}^{2}=\exp \left ( z_{vi}\gamma _{v}\right ) \end{equation*}

A key question to consider regarding heterogeneous inefficiencies is whether heterogeneity in inefficiency exists or if producer-specific inefficiency depends on a set of exogenous factors. As Kim and Schmidt (Reference Kim and Schmidt2008) suggested, one can test for the presence of these factors by regressing $ Y$ on $ X$ and $ Z$ , and then testing the significance of the parameters of these factors using the $ F$ -test.

A summary of the main characteristics of inefficiency models commonly used in the literature is presented in Table 2. A time-variant and time-invariant inefficiency model is derived by omitting $\omega _{i}$ from the four random components inefficiency model. A time-variant inefficiency model is derived by omitting $u_{i0}$ , while a time-invariant inefficiency model is derived by omitting $u_{it}$ from the time-variant and time-invariant inefficiency model.

Table 2. A summary of the main characteristics of inefficiency models

Note: $\omega _{i}$ denotes heterogeneous technologies and NA indicates that the component is not included in the inefficiency model.

4.1.1 Normal-half-normal models

The likelihood function for the normal-half-normal cross-sectional models was derived by Aigner et al. (Reference Aigner, Lovell and Schmidt1977). In these models, it is assumed that inefficiency follows a half-normal distribution $u_{i}\thicksim N^{+}(0,\sigma _{u}^{2})$ ; $f(u)=\left (\sigma _{u}\sqrt {2\pi }\right )^{-1}\exp \left (-u^{2}/2\sigma _{u}^{2}\right )$ , the random errors follow a normal distribution $v_{i}\thicksim N(0,\sigma _{v}^{2})$ , and $u_{i}$ and $v_{i}$ being independently and identically distributed. The likelihood function is defined as the product of the densities of the composed error term $ {\textstyle \prod \nolimits _{i=1}^{N}} f_{\varepsilon }(\varepsilon _{i})$ , where $f_{\varepsilon }(\varepsilon _{i})$ represents the density of the composed error term $\varepsilon _{i}=\nu _{i}-u_{i}$ . Inefficiency can be estimated using Jondrow et al. (Reference Jondrow, Lovell, Materov and Schmidt1982) estimator for the half-normally distributed inefficiency in cross-section models.

\begin{equation*} \widehat {u}_{i}=E(u_{i}\mid \varepsilon _{i})=\sigma _{S}\left [ \frac {\phi \left ( \psi _{i}\right ) }{1-\Phi \left ( \psi _{i}\right ) }-\psi _{i}\right ] \end{equation*}

where $\phi \left ( .\right )$ represents the density of the standard normal distribution, $\Phi \left ( .\right )$ represents the cumulative density function, $\sigma _{S}=\sigma \lambda /\left ( 1+\lambda ^{2}\right ) =\left ( \sigma _{u}^{2}\sigma _{v}^{2}/\sigma ^{2}\right ) ^{1/2}=\sigma _{u}\sigma _{v}/\sigma$ , $\sigma =\left ( \sigma _{u}^{2}+\sigma _{v}^{2}\right ) ^{1/2}$ , $\psi _{i}=\lambda \varepsilon _{i}/\sigma$ , and $\lambda =\sigma _{u}/\sigma _{v}$ . To implement this estimator, it must be evaluated at the estimated parameters $\left ( \widehat {\alpha },\widehat {\beta },\widehat {\sigma }_{u}^{2},\widehat {\sigma }_{v}^{2}\right )$ and the implied values of $\widehat {\lambda }$ , $\widehat {\sigma }^{2}$ , and $\widehat {\varepsilon }_{i}=Y_{i}-\widehat {\alpha }-X_{i}^{^{\prime }}\widehat {\beta }$ . However, Wang and Schmidt (Reference Wang and Schmidt2009) demonstrated that the distribution of $\widehat {E}(u_{i}\mid \varepsilon _{i})$ differs from the distribution of $u_{i}$ unless $\sigma _{v}\rightarrow 0$ . As $\sigma _{v}^{2}$ increases, it converges to $E(u_{i})$ , indicating that $\varepsilon _{i}$ is no longer useful in predicting inefficiency through the conditional mean of the Jondrow et al. (Reference Jondrow, Lovell, Materov and Schmidt1982) estimator. Battese and Coelli (Reference Battese and Coelli1988) proposed an alternative efficiency estimator given by

\begin{equation*} E(TE_{i}\mid \varepsilon _{i})=E\left ( \exp ({-}u_{i})\mid \varepsilon _{i}\right ) =\frac {1-\Phi \left ( \sigma _{S}-\psi _{i}\right ) }{1-\Phi \left ( \psi _{i}\right ) }\exp \left ( \sigma _{S}\psi _{i}+\left ( \left ( \sigma _{S}\right ) ^{2}/2\right ) \right ) \end{equation*}

Fried et al. (Reference Fried, Lovell, Schmidt, Fried, Lovell and Schmidt2008) argued that the efficiency estimator of Battese and Coelli (Reference Battese and Coelli1988) is preferable to $1-E(u_{i}\mid \varepsilon _{i})$ used in the Jondrow et al. (Reference Jondrow, Lovell, Materov and Schmidt1982) estimator because the latter is merely a first-order approximation to the more general infinite power series approximation, exp $\left ( -\left ( u_{i}\mid \varepsilon _{i}\right ) \right ) =1-u_{i}+u_{i}^{2}/2-u_{i}^{3}/3\ldots ..$ However, Fried et al. (Reference Fried, Lovell, Schmidt, Fried, Lovell and Schmidt2008) and Kumbhakar et al. (Reference Kumbhakar, Lien and Hardaker2014) argued that Jondrow et al. (Reference Jondrow, Lovell, Materov and Schmidt1982) and Battese and Coelli (Reference Battese and Coelli1988) estimators are not consistent in cross-sectional models. Although these estimators are unbiased, they do not provide consistent estimates of efficiency, as $p\lim E(u_{i}\mid \varepsilon _{i})-u_{i}\neq 0$ or $E(u_{i}\mid \varepsilon _{i})$ never approaches $u_{i}$ as the number of producers or cross-sectional units approaches infinity.

Greene (Reference Greene1990) argued that the half-normal assumption for the distribution of inefficiency is relatively inflexible and implicitly assumes that most producers are nearly fully efficient. Furthermore, the distribution of the composed error term, $\varepsilon _{i}$ , is no longer normal, as observed by Horrace (Reference Horrace2005). In fact, it might be positively skewed in the wrong direction, leading to full efficiency measures for all producers.Footnote ² Waldman (Reference Waldman1982) demonstrated that if $\varepsilon _{i}$ is incorrectly skewed in the positive direction, then maximum likelihood estimates are equivalent to OLS estimates for $\left ( \alpha, \beta, \sigma _{u}^{2},\sigma _{v}^{2}\right )$ and zero for $\lambda$ . The wrong skewness direction of the OLS composed error term, and consequently a zero maximum likelihood estimate of $\sigma _{u}^{2}$ , is expected given the dependence of the maximum likelihood estimator for $\sigma _{u}^{2}$ on the skewness of the OLS composed error term in the normal-half-normal model.Footnote ³ Feng et al. (Reference Feng, Horrace and Wu2015) suggested using constrained optimization methods to impose the restriction that $\sigma _{u}^{2}\gt 0$ in the normal-half-normal model. Hafner et al. (Reference Hafner, Manner and Simar2018) generalized the inefficiency distribution to allow for the existence of the incorrect skewness, thereby obtaining well defined inefficiency measures.

Pitt and Lee (Reference Pitt and Lee1981), Kumbhakar (Reference Kumbhakar1987), and Battese and Coelli (Reference Battese and Coelli1988) extended the normal-half-normal model proposed by Aigner et al. (Reference Aigner, Lovell and Schmidt1977) to the panel data time-invariant inefficiency model. Inefficiency is estimated using an extension of the Jondrow et al. (Reference Jondrow, Lovell, Materov and Schmidt1982) estimator for the panel data model.

(2) \begin{equation} \widehat {u}_{i}=E(u_{i}\mid \varepsilon _{it})=\varphi _{i}^{N}+\sigma _{P}\left [ \frac {\phi \left ( \varphi _{i}^{N}/\sigma _{P}\right ) }{1-\Phi \left ( \varphi _{i}^{N}/\sigma _{P}\right ) }\right ] \end{equation}

where $\varphi _{i}^{N}=\left ( -\sigma _{u}^{2}{\textstyle \sum \nolimits _{t=1}^{T}} \varepsilon _{it}\right ) /\left ( \sigma _{v}^{2}+T\sigma _{u}^{2}\right )$ , and $\sigma _{P}=\left ( \sigma _{v}^{2}\sigma _{u}^{2}\right ) /\left ( \sigma _{v}^{2}+T\sigma _{u}^{2}\right )$ . Kumbhakar (Reference Kumbhakar1987) noted that these estimates are asymptotically consistent. Lee (Reference Lee1996) introduced a tail-truncated half-normal distribution to incorporate a bound for inefficiency, as the extremely inefficient producers cannot survive in highly competitive markets. He introduced the threshold parameter of the minimum efficiency for survival, denoted as $\theta$ , where $u_{i}\thicksim N^{+}(0,\sigma _{u}^{2})$ , $0\leq u_{i}\leq \theta$ . Thus, the variance of inefficiency depends on two parameters: $\sigma _{u}^{2}$ and $\theta$ .

4.1.2 Normal-exponential models

Aigner et al. (Reference Aigner, Lovell and Schmidt1977) and Meeusen and van den Broeck (Reference Meeusen and Van den Broeck1977) proposed a likelihood function under the assumption that $u_{i}$ follows an exponential distribution; $f(u)=\theta \exp \left ( -\theta u\right )$ , where $\theta \gt 0$ and $\theta =\sigma _{u}^{-1}$ . Additionally, $v_{i}$ is assumed to follow a normal distribution, $v_{i}\thicksim N(0,\sigma _{v}^{2})$ . Inefficiency can be estimated using the Jondrow et al. (Reference Jondrow, Lovell, Materov and Schmidt1982) estimator for exponentially distributed inefficiency in cross-sectional models.

\begin{equation*} \widehat {u}_{i}=E(u_{i}\mid \varepsilon _{i})=\sigma _{v}\left [ \frac {\phi \left ( \varphi _{i}^{E}\right ) }{\Phi \left ( \varphi _{i}^{E}\right ) }+\varphi _{i}^{E}\right ] \end{equation*}

where $\varphi _{i}^{E}=\left ( \varepsilon _{i}\sigma _{u}-\sigma _{v}^{2}\right ) /\left ( \sigma _{u}\sigma _{v}\right )$ . Kim and Schmidt (Reference Kim and Schmidt2000) extended the normal-exponential model proposed by Aigner et al. (Reference Aigner, Lovell and Schmidt1977) to a panel data time-invariant inefficiency model. Inefficiency is estimated using an extension of the Jondrow et al. (Reference Jondrow, Lovell, Materov and Schmidt1982) estimator for the panel data model by replacing $\varepsilon _{i}$ by $\overline {\varepsilon }_{i}$ and $\sigma _{v}^{2}$ by $\sigma _{v}^{2}/T$ . A simulation study conducted by Horrace and Parmeter (Reference Horrace and Parmeter2018) indicates that a Laplace model, where $ v_{i}$ follows a Laplace distribution and $ u_{i}$ follows a truncated Laplace distribution, performs relatively well compared to the normal-exponential model when $ v_{i}$ is misspecified.

4.1.3 Normal-Gamma models

Greene (Reference Greene1980a, Reference Greene1980b), Stevenson (Reference Stevenson1980), and Greene (Reference Greene1990) assumed a gamma distribution for the inefficiency term, where $f(u)=\left [\theta ^{P}/\Gamma (P)\right ] \exp \left ({-}\theta u\right )u^{P-1}$ , $P\gt 0$ , $\theta =\sigma _{u}^{-1}$ , $\Gamma (P)= {\textstyle \int \nolimits _{0}^{\infty }}t^{P-1}e^{-t}dt$ , and $v_{i}\thicksim N(0,\sigma _{v}^{2})$ . Stevenson (Reference Stevenson1980) considered only the Erlang form (integer values of $P$ ; 1.0 and 2.0), which produces a tractable formulation for $f_{\varepsilon }(\varepsilon _{i})$ but significantly restricts the model. Beckers and Hammond (Reference Beckers and Hammond1987) derived the log-likelihood function for $f_{\varepsilon }(\varepsilon _{i})$ without limiting $P$ to integer values; however, the resulting functional form was intractable. When $P=1$ , the normal-gamma model reverts to the normal-exponential model. The inefficiency estimator for the gamma model is

\begin{equation*} \widehat {u}_{i}=E(u_{it}\mid \varepsilon _{it})=\frac {q\left ( P,\varepsilon _{it}\right ) }{q\left ( P-1,\varepsilon _{it}\right ) }\end{equation*}

The normal-gamma distribution provides a more flexible parameterization of the distribution. However, the computational complexity of the maximum likelihood estimator limits its application in empirical studies. Various efforts, including those by Ritter and Simar (Reference Ritter and Simar1997) and Greene (Reference Greene2003), have been made to simplify the computation using simulation methods.

4.1.4 Normal-truncated-normal models

Stevenson (Reference Stevenson1980) argued that the assumption of a zero mean in the Aigner et al. (Reference Aigner, Lovell and Schmidt1977) model is an unnecessary constraint. He suggested that inefficiency follows a non-negative truncated distribution, represented as $u_{i}\thicksim N(\mu, \sigma _{u}^{2})$ ; $f(u)=\left ( \Phi \left ( \mu /\sigma _{u}\right ) \sigma _{u}\sqrt {2\pi }\right ) ^{-1}\exp \left ( -\left ( u-\mu \right ) ^{2}/2\sigma _{u}^{2}\right )$ . Greene (Reference Greene and Greene1993) demonstrated that the conditional expectation of inefficiency for the truncated-normal distribution, where $\mu$ can vary from zero in either direction, is obtained by replacing $\psi _{i}$ with $\psi _{i}^{T}=\lambda \varepsilon _{i}/\sigma +\mu /\lambda \sigma$ .

Pitt and Lee (Reference Pitt and Lee1981) expanded the normal-truncated-normal model to accommodate panel data with time-invariant inefficiency. Battese and Coelli (Reference Battese and Coelli1988) and Battese et al. (Reference Battese, Coelli and Colby1989) further developed the Jondrow et al. (Reference Jondrow, Lovell, Materov and Schmidt1982) estimator for application in the panel data model.

\begin{equation*} \widehat {u}_{i}=E(u_{i}\mid \varepsilon _{it})=\varphi _{i}^{T}+\sigma _{P}\left [ \frac {\phi \left ( \varphi _{i}^{T}/\sigma _{P}\right ) }{1-\Phi \left ( \varphi _{i}^{T}/\sigma _{P}\right ) }\right ] \end{equation*}

where $\varphi _{i}^{T}=\left ( \mu \sigma _{v}^{2}-\sigma _{u}^{2}{\textstyle \sum \nolimits _{t=1}^{T}} \varepsilon _{it}\right ) /\left ( \sigma _{v}^{2}+T\sigma _{u}^{2}\right )$ . By setting $\mu = 0$ , it reverts to the estimator for the normal-half-normal model in equation (2). Battese and Coelli (Reference Battese and Coelli1988, Reference Battese and Coelli1992) extended the panel data to $E\left ( \exp ({-}u_{i})\mid \varepsilon _{i}\right )$ as

\begin{equation*} E\left ( \exp ({-}u_{i})\mid \varepsilon _{it}\right ) =\left [ \frac {\Phi \left [ \left ( \varphi _{i}^{T}/\sigma _{P}\right ) -\sigma _{P}\right ] }{\Phi \left ( \varphi _{i}^{T}/\sigma _{P}\right ) }\right ] \exp \left ( -\varphi _{i}^{T}+\left ( \sigma _{P}/2\right ) \right ) \end{equation*}

Almanidis et al. (Reference Almanidis, Qian, Sickles, Almanidis, Qian and Sickles2014) specified inefficiency as a doubly truncated-normal distribution. In addition to the zero lower bound, they specified an upper bound for inefficiency to exclude extremely inefficient producers. The upper bound, $\theta _{t}$ , is assumed to be the sum of weighted polynomials, $\theta _{t}={\textstyle \sum \nolimits _{i=0}^{N}}b_{i}\left ( t/T\right )^{i}$ , where $t=1,\ldots ..,T$ , and $b_{i}$ are constants. Their specification provides a closed-form solution for $f_{\varepsilon }(\varepsilon _{i})$ and the log-likelihood. Furthermore, this specification results in non-zero estimates of $\sigma _{u}^{2}$ in the presence of wrong skewness of the composed error term.

The truncated-normal distribution is applicable when producers are assumed to be inefficient, as it has a mode at zero only if $ \mu \leq 0$ . Furthermore, it provides a method for introducing heterogeneity into the distribution of inefficiency by incorporating characteristics specific to producers into the mean, variance, or both parameters of the inefficiency distribution.

4.1.5 Skew-normal models

Recent studies have focussed on the distribution of the composed error term $ f_{\varepsilon }(\varepsilon _{i})$ rather than on the distribution of inefficiency. In models with four random components, these components $ \left (\omega _{i}+\nu _{it}-u_{i0}-u_{it}\right )$ can be treated as two terms because they can be expressed as the sum of the time-invariant terms $ \left ( \xi _{i}=\omega _{i}-u_{i0}\right )$ and the time-variant terms $ \left (\varepsilon _{it}=\nu _{it}-u_{it}\right )$ . The time-invariant terms combine the producer-specific effects for unobserved heterogeneity $(\omega _{i})$ and time-invariant inefficiency $(u_{i0})$ , whereas the time-variant terms combine the random errors $(\nu _{it})$ and time-variant inefficiency $(u_{it})$ . These two terms are assumed to result from the difference between a normal random variable and an independent normal random variable that is left-truncated at zero. Consequently, each of the two terms follows its own skew-normal distribution rather than a normal distribution.Footnote ⁴

The full unconditional log-likelihood function for this model, based on the joint distribution of $\left ( \varepsilon _{it},\xi _{i}\right )$ , was derived by Colombi et al. (Reference Colombi, Kumbhakar, Martini and Vittadini2014). They estimated the four random components as $E\left (\exp (\omega _{i})\mid y_{i}\right )$ , and $E\left (\exp (t^{^{\prime }}u_{i})\mid y_{i}\right )$ , where the first element of $E\left (\exp (t^{^{\prime }}u_{i})\mid y_{i}\right )$ represents the conditional expected value of the time-invariant inefficiency $u_{i0}$ for each producer $i$ . However, the computational complexity of the maximum likelihood estimator arises from the $(T+1)$ dimentional multivariate normal integrals.Footnote ⁵ Tsionas and Kumbhakar (Reference Tsionas and Kumbhakar2014) observed that the maximum likelihood estimator developed by Colombi et al. (Reference Colombi, Kumbhakar, Martini and Vittadini2014) becomes computationally challenging when $ T$ is large. However, for models with time-invariant inefficiency, whether or not they include a producer-specific component to account for heterogeneous technologies, the integral is one-dimensional. For models with time-variant inefficiency that do not include a producer-specific component for heterogeneous technologies, it is a product of $ T$ one-dimensional integrals. Consequently, the computational challenge is primarily associated with time-variant inefficiency models that include a producer-specific component for heterogeneous technologies, or time-invariant and time-variant inefficiency models, where time-variant inefficiency coexists with time-invariant inefficiency. Fillipini and Greene (Reference Filippini and Greene2016) utilized Butler and Moffitt’s (Reference Butler and Moffitt1982) formulation to propose a simplified density of $ y_{i}$ conditional on $ u_{i0}$ and $ \omega _{i}$ , which is the product over time of $ T$ univariate closed skew-normal densities.

The efficiency estimates are derived using the results from Colombi et al. (Reference Colombi, Kumbhakar, Martini and Vittadini2014), which are based on the moment-generating function for the closed skew-normal distribution. Inefficiency is calculated using the expression $-\log E\left ( \exp (t^{^{\prime }}u_{i})\mid e_{i}\right )$ , computed element by element. Fillipini and Greene (Reference Filippini and Greene2016) followed the methodologies of Kumbhakar and Heshmati (Reference Kumbhakar and Heshmati1995) and Kumbhakar et al. (Reference Kumbhakar, Lien and Hardaker2014) to measure total efficiency as follows:

\begin{equation*} \text {Total efficiency}_{it}=E\left ( \exp ({-}u_{i0})\mid e_{i}\right ) \times E\left ( \exp ({-}u_{it})\mid e_{i}\right ) \end{equation*}

Concerning the significance of distribution assumptions, if the goal is to estimate inefficiency specific to individual producers, then the assumption about the distribution of inefficiency becomes crucial. However, if the objective is to compare the rankings of producers, employing models without distribution assumptions or adhering to the suggestion by Ritter and Simar (Reference Ritter and Simar1997) to utilize a simple one-parameter distribution for inefficiency may be adequate.

4.1.6 Confidence intervals

Distributions imposed on $v$ and $u$ create distributions for $\left (u\mid \varepsilon \right )$ and $\left ( \exp ({-}u)\mid \varepsilon \right )$ , which can be used to construct confidence intervals for inefficiency. Several studies have demonstrated that it is possible to obtain confidence intervals for any of the inefficiency estimators. Hjalmarsson et al. (Reference Hjalmarsson, Kumbhakar and Heshmati1996) developed confidence intervals for the Jondrow et al. (Reference Jondrow, Lovell, Materov and Schmidt1982) estimator, and Bera and Sharma (Reference Bera and Sharma1999) did so for the Battesse and Coelli (Reference Battese and Coelli1988) estimator. Horrace and Schmidt (Reference Horrace and Schmidt1996) derived lower and upper bounds on $\left (\exp ({-}u)\mid \varepsilon \right )$ based on the lower and upper bounds of $\left (u\mid \varepsilon \right )$ . However, Wheat et al. (Reference Wheat, Greene and Smith2014) argued that the form of confidence intervals derived by Horrace and Schmidt (Reference Horrace and Schmidt1996) is not of minimum width because $f\left ( u\mid \varepsilon \right )$ is truncated-normal at zero and thus asymmetric. They proposed a minimum width prediction interval for $u$ given $\varepsilon$ . Parmeter and Kumbhakar (Reference Parmeter and Kumbhakar2014) noted that the narrower interval proposed by Wheat et al. (Reference Wheat, Greene and Smith2014) is preferable to the intervals of Horrace and Schmidt (Reference Horrace and Schmidt1996) if the aim is to accurately predict producer-specific inefficiency.

4.2.1 The prior distribution

The prior distributions of the parameters to be estimated, including inefficiency, represent information that is not contained in the data and are expressed as a probability distribution, $p\left (\theta \right )$ . These distributions are categorized as either informative priors, which are based on previous findings and theoretical predictions, or uninformative priors, which are based on a lack of prior knowledge available for estimation.

Specifying a uniform or flat prior distribution allows the prior to play a minimal role in estimating the posterior distribution by relying on the data through the likelihood function. This approach is equivalent to specifying a prior distribution with a large variance, which makes the prior distribution of the parameter values nearly flat. However, non-informative prior distributions are often improper. Fernandez et al. (Reference Fernandez, Osiewalski and Steel1997) demonstrated that choosing an uninformative prior on the scale parameter leads to an improper prior.

Informative priors convey information and summarize existing knowledge about parameters. Since the normal distribution allows for negative numbers, it is not suitable as a prior distribution for inefficiency or scale parameters. Van den Broeck et al. (Reference Van den Broeck, Koop, Osiewalski and Steel1994) found that the exponential distribution is more robust to prior assumptions than other distributions. Alvarez et al. (Reference Alvarez, Niemi, Simpson, Alvarez, Niemi and Simpson2014) compared the inverse Wishart, scaled inverse Wishart, and hierarchical inverse Wishart as potential priors for the scale parameter in multivariate models. They found that all priors perform well except the inverse Wishart prior, which is biased toward large values when the true variance is small relative to the prior mean. Esheba and Serletis (Reference Esheba and Serletis2023) utilized the Wishart distribution for the scale parameter and found it to be biased toward large values, resulting in large values for the scale parameter and consequently large values for the inefficiency measures. They also noted that the MCMC algorithm for a system of equations terminated after a small number of iterations due to the large values involved. In general, the prior distribution for the scale parameter, which plays a crucial role in the estimation of inefficiency, is essential in any multivariate model and becomes more challenging as the dimension increases due to the quadratic growth in the number of parameters and the need to ensure the matrix remains non-negative definite.

Informative priors can be utilized to impose constraints derived from economic theory, such as monotonicity and curvature constraints, as noted in Terrell (Reference Terrell1996). They can also be employed for linear constraints among the elements of the parameters, as demonstrated by Geweke (Reference Geweke1993), or for constraints on inefficiency, where $ u \geq 0$ , as shown by Feng et al. (Reference Feng, Wang and Serletis2018) and Esheba and Serletis (Reference Esheba and Serletis2023). It is important to note that selecting a prior distribution that is conjugate to the likelihood results in a posterior that retains the same form as the prior.

4.2.2 The posterior distribution

Updating the prior information of the parameters is accomplished by combining the prior distribution, $ p(\theta )$ , with the likelihood function, $ L(Y, \theta )$ . This process results in the posterior distribution, which serves as the basis for Bayesian estimation and is defined by Bayes’ theorem as follows:

\begin{equation*} p(\theta \mid Y)\propto L(Y,\theta )p\left ( \theta \right ) \end{equation*}

In this context, $ p(\theta \mid Y)$ represents the posterior distribution, which is proportional to the likelihood function multiplied by the prior. The posterior mean $ E(\theta \mid Y)$ serves as the optimal Bayesian estimator of $ \theta$ . However, when the model involves multidimensional parameters to be estimated, the posterior distribution becomes a joint posterior distribution. The marginal posterior distribution for a given parameter $ \theta _{i}$ is defined by integrating the joint posterior density of $ \theta$ with respect to all elements of $ \theta$ other than $ \theta _{i}$ . This process can be too complex for direct analytical integration or may not be analytically tractable. Implementing the Bayesian approach often requires the use of an iterative MCMC algorithm. Two common algorithms are Gibbs sampling, introduced by Geman and Geman (Reference Geman and Geman1984), and the MetropolisHastings algorithm, introduced by Metropolis et al. (Reference Metropolis, Rosenbluth, Rosenbluth, Teller and Teller1953) and further developed by Hastings (Reference Hastings1970).

When the joint posterior distribution is challenging to work with, Gibbs sampling, which involves drawing from the conditional posterior distributions, can be employed to approximate joint and marginal distributionsFootnote ⁶ This method is advantageous in situations where the conditional posterior distributions have relatively simpler forms than the joint distribution, facilitating simulation. To assess whether the draws from the conditional posterior distributions have converged to the marginal posterior distribution, Geweke (Reference Geweke and Geweke1992) proposed a convergence test. If there is insufficient evidence for convergence, the number of draws must be increased.

The Gibbs sampling algorithm relies on conditional distributions. However, in some situations, conditional posterior distributions do not belong to any known family distributions or are not available in closed form, making simulation from them challenging. In such situations, the Metropolis-Hastings algorithm, which is more general than Gibbs sampling, serves as an alternative MCMC algorithm that can be used to approximate the posterior distribution. The Metropolis-Hastings algorithm requires the specification of a proposal density, $ q(\theta ^{\ast } \mid \theta ^{S})$ , which is easier to simulate from than the target density, $ p(\theta \mid Y)$ .

The initial MCMC iterations are discarded as a burn-in, and estimates of the parameters are obtained by averaging over the remaining iterations. It should be noted that the Gibbs sampler is considered a specific case of the Metropolis-Hastings algorithm, where the candidate density $ q(\theta ^{\ast } \mid \theta ^{S})$ coincides with the target density, resulting in an acceptance probability of 1 for each draw.

5.1.1 The translog functional form

The translog functional form is a generalization of the Cobb-Douglas functional form and was introduced by Christensen et al. (Reference Christensen, Jorgenson and Lau1973). It is a locally flexible functional form that provides a second-order local approximation. Caves and Christensen (Reference Caves and Christensen1980), Guilkey and Lovell (Reference Guilkey and Lovell1980), Barnett and Lee (Reference Barnett and Lee1985), and Barnett et al. (Reference Barnett, Lee and Wolfe1985) argued that most locally flexible functional forms are not globally regular and have very small regions where theoretical regularity conditions are satisfied. The translog functional form is defined over $ N$ inputs and $ M$ outputs as follows:

\begin{align*} \ln \left ( D\left ( x,y\right ) \right ) & =\alpha _{0}+\sum _{n=1}^{N}\alpha _{n}\ln x_{n}+\sum _{m=1}^{M}\beta _{m}\ln y_{m}+\frac {1}{2}\sum _{n=1}^{N}\sum _{n^{\prime }=1}^{N}\alpha _{nn^{\prime }}\ln x_{n}\ln x_{n^{\prime }}\\ & +\frac {1}{2}\sum _{m=1}^{M}\sum _{m^{\prime }=1}^{M}\beta _{mm^{\prime }}\ln y_{m}\ln y_{m^{\prime }}+\sum _{n=1}^{N}\sum _{m=1}^{M}\gamma _{nm}\ln x_{n}\ln y_{m} \end{align*}

Symmetry requires that $\alpha _{nn^{\prime }}=\alpha _{n^{\prime }n}$ $(n\neq n^{\prime })$ , and $\beta _{mm^{\prime }}=\beta _{m^{\prime }m}$ $(m\neq m^{\prime })$ . The translog functional form includes numerous parameters that require estimation, totaling $\left ( k^{2}+3k+2\right ) /2$ parameters, including the intercept. It is linear in the parameters, which can be constrained to satisfy the homogeneity property of standard and hyperbolic distance functions; however, it cannot be constrained to satisfy the translation property of directional distance functions.

The constraints necessary for achieving homogeneity of degree one in inputs are: ${\textstyle \sum \nolimits _{n=1}^{N}}\alpha _{n}=1$ , ${\textstyle \sum \nolimits _{n^{\prime }=1}^{N}}\alpha _{nn^{\prime }}=0$ , and ${\textstyle \sum \nolimits _{n=1}^{N}}\gamma _{nm}=0$ . One approach to imposing these restrictions on the input distance function is to normalize the function by one of the inputs. This is achieved by setting the parameter of the homogeneity property to $\lambda =1/x_{N}$ , resulting in $D_{I}\left (y,x/x_{N}\right ) =D_{I}\left (y,x\right )/x_{N}$ . For further details, refer to Sturm and Williams (Reference Sturm and Williams2008). The constraints necessary for achieving homogeneity of degree one in outputs are: ${\textstyle \sum \nolimits _{m=1}^{M}}\beta _{m}=1$ , $ {\textstyle \sum \nolimits _{m^{\prime }=1}^{M}}\beta _{mm^{\prime }}=0$ , and ${\textstyle \sum \nolimits _{m=1}^{M}}\gamma _{nm}=0$ . One method to impose these restrictions on the output distance function is to normalize the function by one of the outputs. This is achieved by setting the parameter of the homogeneity property to $\lambda =1/y_{M}$ , resulting in $D_{O}\left (x,y/y_{M}\right ) =D_{O}\left (x,y\right )/y_{M}$ . For further details, refer to O’Donnell and Coelli (Reference O’Donnell and Coelli2005). The restrictions required for almost homogeneity of degrees $-1$ , $1$ , and $1$ are: ${\textstyle \sum \nolimits _{m=1}^{M}}\beta _{m}-{\textstyle \sum \nolimits _{n=1}^{N}}\alpha _{n}=1$ , ${\textstyle \sum \nolimits _{m=1}^{M}}\gamma _{nm}-{\textstyle \sum \nolimits _{n^{\prime }=1}^{N}} \alpha _{nn^{\prime }}=0$ , and ${\textstyle \sum \nolimits _{m^{\prime }=1}^{M}}\beta _{mm^{\prime }}-{\textstyle \sum \nolimits _{n=1}^{N}} \gamma _{nm}=0$ . One approach to imposing these restrictions on the hyperbolic distance function is to normalize the function by one of the inputs. This is achieved by setting the parameter of the homogeneity property to $\lambda =1/x_{N}$ , resulting in $D_{H}\left ( x/x_{N},yx_{N}\right )=D_{H}\left ( x,y\right ) /x_{N}$ . Alternatively, normalization can be accomplished by one of the outputs by setting $\lambda =1/y_{M}$ , resulting in $D_{H}\left (xy_{M},y/y_{M}\right )=D_{H}\left (x,y\right )/y_{M}$ . For further details, refer to Cuesta and Zofio (Reference Cuesta and Zofio2005). The first-order and second-order partial derivatives are expressed as follows:

\begin{equation*} \frac {\partial \ln \left ( D\left ( x,y\right ) \right ) }{\partial \ln x_{n}}=\alpha _{n}+\sum _{n^{\prime }=1}^{N}\alpha _{nn^{\prime }}\ln x_{n^{\prime }}+\sum _{m=1}^{M}\gamma _{nm}\ln y_{m}, \ \ \ \ \ \ \frac {\partial ^{2}\ln \left ( D\left ( x,y\right ) \right ) }{\partial \ln x_{n}\ln x_{n^{\prime }}}=\alpha _{nn^{\prime }}\end{equation*}

\begin{equation*} \frac {\partial \ln \left ( D\left ( x,y\right ) \right ) }{\partial \ln y_{m}}=\beta _{m}+\sum _{m^{\prime }=1}^{M}\beta _{mm^{\prime }}\ln y_{m^{\prime }}+\sum _{n=1}^{N}\gamma _{nm}\ln x_{n}, \ \ \ \ \ \ \frac {\partial ^{2}\ln \left ( D\left ( x,y\right ) \right ) }{\partial \ln y_{m}\ln y_{m^{\prime }}}=\beta _{mm^{\prime }}\end{equation*}

The translog functional form, unlike the Cobb-Douglas functional form, is neither monotonic nor globally convex. Caves and Christensen (Reference Caves and Christensen1980) noted that the translog functional form exhibits satisfactory local properties when the technology is nearly homothetic and the substitution between factors of production is high. Guilkey et al. (Reference Guilkey, Lovell and Sickles1983) demonstrated that the translog functional form is globally regular if and only if the technology is Cobb-Douglas. Färe and Vardanyan (Reference Färe and Vardanyan2016) found that the translog functional form often violates theoretical regularity conditions and requires the imposition of appropriate regularity conditions, which significantly compromise its flexibility. Their findings align with the simulation results of Wales (Reference Wales1977) and Guilkey et al. (Reference Guilkey, Lovell and Sickles1983), who compared the performance of various functional forms, including the translog.

5.1.2 The quadratic functional form

Chambers (Reference Chambers and Chambers1998) suggested the use of a quadratic functional form for directional distance functions, as its parameters can be constrained to satisfy the translation property. Lau (Reference Lau and Lau1978b) introduced this quadratic functional form, which is expressed as follows:

(3) \begin{align} D\left ( x,y\right ) & =\alpha _{0}+\sum _{n=1}^{N}\alpha _{n}x_{n}+\sum _{m=1}^{M}\beta _{m}y_{m}+\frac {1}{2}\sum _{n=1}^{N}\sum _{n^{\prime }=1} ^{N}\alpha _{nn^{\prime }}x_{n}x_{n^{\prime }}\nonumber \\ & +\frac {1}{2}\sum _{m=1}^{M}\sum _{m^{\prime }=1}^{M}\beta _{mm^{\prime }} y_{m}y_{m^{\prime }}+\sum _{n=1}^{N}\sum _{m=1}^{M}\gamma _{nm}x_{n}y_{m} \end{align}

Symmetry requires that $\alpha _{nn^{\prime }}=\alpha _{n^{\prime }n}$ $(n\neq n^{\prime })$ , and $\beta _{mm^{\prime }}=\beta _{m^{\prime }m}$ $(m\neq m^{\prime })$ . The quadratic functional form includes numerous parameters that require estimation, totaling $\left ( k^{2}+3k+2\right )/2$ parameters, including the intercept. It is linear in the parameters, which can be restricted to satisfy the translation property of the directional distance functions. The set of linear restrictions required for the translation property is as follows: ${\textstyle \sum \nolimits _{m=1}^{M}}\beta _{m}g_{y_{m}}-{\textstyle \sum \nolimits _{n=1}^{N}}\alpha _{n}g_{x_{n}}=-1$ ; ${\textstyle \sum \nolimits _{m=1}^{M}}\beta _{mm^{\prime }}g_{y_{m}}=0$ ; ${\textstyle \sum \nolimits _{n=1}^{N}}\alpha _{nn^{\prime }}g_{x_{n}}=0$ ; ${\textstyle \sum \nolimits _{m=1}^{M}}\gamma _{nm}g_{y_{m}}=0$ ; and $ {\textstyle \sum \nolimits _{n=1}^{N}}\gamma _{nm}g_{x_{n}}=0$ . One approach to imposing these restrictions is to apply them directly in Equation (3) to derive a restricted version, as illustrated by Atkinson and Tsionas (Reference Atkinson and Tsionas2016). Alternatively, these restrictions can be imposed by setting the parameter of the translation property, $\alpha$ , equal to a selected input, $\alpha =x_{N}$ , or the negative of a selected output, $\alpha =-y_{M}$ . The corresponding direction vector is then normalized so that $g_{x_{N}}=1$ or $g_{y_{M}}=1$ . In the situation where $\alpha =x_{N}$ is chosen, the expression $\vec {D}_{T}\left ( \widetilde {x}-x_{N}\widetilde {g}_{x},y+x_{N}g_{y};\ g_{x},g_{y}\right ) =\vec {D}_{T}\left (x,y;\ g_{x},g_{y}\right )-x_{N}$ holds, where $\widetilde {x}=(x_{1},\ldots ..,x_{N-1})$ and $\widetilde {g}_{x}=(g_{x_{1}},\ldots .,g_{x_{N-1}})$ . The input $x_{N}$ is absent from $\vec {D}_{T}\left ( \widetilde {x}-x_{N}\widetilde {g}_{x},y+x_{N}g_{y};\ g_{x},g_{y}\right )$ because $x_{N}-x_{N}(1)=0$ . In the situation where $\alpha =-y_{M}$ is chosen, the expression $\vec {D}_{T}\left ( x+y_{M}g_{x},\widetilde {y}-y_{M}\widetilde {g}_{y};\ g_{x},g_{y}\right ) =\vec {D}_{T}\left (x,y;\ g_{x},g_{y}\right ) +y_{M}$ holds, where $\widetilde {y}=(y_{1},\ldots ..,y_{M-1})$ and $\widetilde {g}_{y}=(g_{y_{1}},\ldots .,g_{y_{M-1}})$ . The output $y_{M}$ is absent from $\vec {D}_{T}\left ( x+y_{M}g_{x},\widetilde {y}-y_{M} \widetilde {g}_{y};\ g_{x},g_{y}\right )$ because $y_{M}-y_{M}(1)=0$ ; see, for example, Malikov et al. (Reference Malikov, Kumbhakar and Tsionas2016). The first-order and second-order partial derivatives are expressed as follows:

\begin{equation*} \frac {\partial D\left ( x,y\right ) }{\partial x_{n}}=\alpha _{n}+\sum _{n^{\prime }=1}^{N}\alpha _{nn^{\prime }}x_{n^{\prime }}+\sum _{m=1}^{M}\gamma _{nm}y_{m}, \ \ \ \ \ \ \frac {\partial ^{2}D\left ( x,y\right ) }{\partial x_{n}x_{n^{\prime }}}=\alpha _{nn^{\prime }}\end{equation*}

\begin{equation*} \frac {\partial D\left ( x,y\right ) }{\partial y_{m}}=\beta _{m}+\sum _{m^{\prime }=1}^{M}\beta _{mm^{\prime }}y_{m^{\prime }}+\sum _{n=1}^{N}\gamma _{nm}x_{n}, \ \ \ \ \ \ \frac {\partial ^{2}D\left ( x,y\right ) }{\partial y_{m}y_{m^{\prime }}}=\beta _{mm^{\prime }}\end{equation*}

Thompson (Reference Thompson1988) noted that the quadratic functional form can satisfy global curvature restrictions without additional constraints in estimation. This finding is supported by the results of Färe and Vardanyan (Reference Färe and Vardanyan2016), who demonstrated that the quadratic functional form satisfies global regularity without curvature restrictions and maintains its flexibility. A simulation study by Chambers et al. (Reference Chambers, Färe, Grosskopf and Vardanyan2013) suggests that the quadratic functional form outperforms the translog in large samples with a relatively high degree of curvature. Diewert and Fox (Reference Diewert and Fox2008) noted that curvature restrictions can be globally imposed on the quadratic functional form without losing flexibility. However, monotonicity cannot be imposed simultaneously with curvature without compromising the flexibility of the functional form. As Barnett (Reference Barnett2002) noted, imposing global curvature on the quadratic functional form may induce spurious violations of monotonicity.

5.1.3 The logarithmic-transcendental functional form

Chambers (Reference Chambers and Chambers1998) suggested the use of a logarithmic-transcendental functional form for directional distance functions due to its inherent satisfaction of the translation property. However, it has been largely ignored in the inefficiency literature, as researchers have favored the quadratic functional form for its linearity in parameters. The logarithmic-transcendental or the transcendental-exponential functional form is a flexible functional form that provides a second-order approximation and can be expressed as follows:

(4) \begin{align} \exp \left ( D\left ( x,y\right ) \right ) & =\alpha _{0}+\frac {1}{2}\sum _{n=1}^{N}\sum _{n^{\prime }=1}^{N}\alpha _{nn^{\prime }}\exp \left ( \frac {x_{n}}{2}\right ) \exp \left ( \frac {x_{n^{\prime }}}{2}\right ) \nonumber \\ & +\frac {1}{2}\sum _{m=1}^{M}\sum _{m^{\prime }=1}^{M}\beta _{mm^{\prime }}\exp \left ( -\frac {y_{m}}{2}\right ) \exp \left ( -\frac {y_{m^{\prime }}}{2}\right ) +\sum _{n=1}^{N}\sum _{m=1}^{M}\gamma _{nm}\exp \left ( \frac {x_{n}}{2}\right ) \exp \left ( -\frac {y_{m}}{2}\right ) \end{align}

The logarithmic-transcendental functional form requires fewer parameters to be estimated than the translog and quadratic functional forms, with a total of $\left ( k^{2}+k+2\right ) /2$ parameters, including the intercept. Symmetry requires that $\alpha _{nn^{\prime }}=\alpha _{n^{\prime }n}$ $(n\neq n^{\prime })$ , and $\beta _{mm^{\prime }}=\beta _{m^{\prime }m}$ $(m\neq m^{\prime })$ . The first-order partial derivatives are expressed as follows:

\begin{equation*} \frac {\partial \exp \left ( D\left ( x,y\right ) \right ) }{\partial \exp \left ( \frac {x_{n}}{2}\right ) }=\sum _{n^{\prime }=1}^{N}\alpha _{nn^{\prime }}\exp \left ( \frac {x_{n^{\prime }}}{2}\right ) +\sum _{m=1}^{M}\gamma _{nm}\exp \left ( -\frac {y_{m}}{2}\right ) \end{equation*}

\begin{equation*} \frac {\partial \exp \left ( D\left ( x,y\right ) \right ) }{\partial \exp \left ( -\frac {y_{m}}{2}\right ) }=\sum _{m^{\prime }=1}^{M}\beta _{mm^{\prime }}\exp \left ( -\frac {y_{m^{\prime }}}{2}\right ) +\sum _{n=1}^{N}\gamma _{nm}\exp \left ( \frac {x_{n}}{2}\right ) \end{equation*}

The second-order partial derivatives are expressed as follows:

\begin{equation*} \frac {\partial ^{2}\exp \left ( D\left ( x,y\right ) \right ) }{\partial \exp \left ( \frac {x_{n}}{2}\right ) \exp \left ( \frac {x_{n^{\prime }}}{2}\right ) }=\alpha _{nn^{\prime }}\end{equation*}

\begin{equation*} \frac {\partial ^{2}\exp \left ( D\left ( x,y\right ) \right ) }{\partial \exp \left ( -\frac {y_{m}}{2}\right ) \exp \left ( -\frac {y_{m^{\prime }}}{2}\right ) }=\beta _{mm^{\prime }}\end{equation*}

Empirical techniques can also be used to assess the ability of different functional forms to approximate the unknown underlying function. Several methods have been proposed in the literature, including Monte Carlo simulations, parametric modeling, and constructive techniques. Monte Carlo simulations assess the approximation capabilities of different functional forms relative to the underlying technology. For applications of this technique, see, for example, Guilkey and Lovell (Reference Guilkey and Lovell1980), Giannakas et al. (Reference Giannakas, Tran and Tzouvelekas2003b), Färe et al. (Reference Färe, Martins-Filho and Vardanyan2010), Chambers et al. (Reference Chambers, Färe, Grosskopf and Vardanyan2013), and Färe and Vardanyan (Reference Färe and Vardanyan2016). Parametric modeling examines the plausibility of various functional forms in fitting actual data. For applications of this technique, see, for example, Griffin et al. (Reference Griffin, Montgomery and Rister1987), Giannakas, et al. (Reference Giannakas, Tran and Tzouvelekas2003a), and Feng and Serletis (Reference Feng and Serletis2008). The primary challenge with parametric modeling is that the true functional form for production technology is unknown. Evaluating the performance of different functional forms in fitting actual data is beneficial if the focus is on analyzing the data itself, rather than the functional forms. As noted by Giannakas, et al. (Reference Giannakas, Tran and Tzouvelekas2003a), the appropriate functional form in this context is specific to each situation. Constructive techniques provide a method to determine preferable functional forms by deriving and graphically displaying their regular regions. For applications of this technique, see, for example, Caves and Christensen (Reference Caves and Christensen1980), and Barnett et al. (Reference Barnett, Lee and Wolfe1985, Reference Barnett, Lee and Wolfe1987).

5.2.1 Heterogeneity in the production technology

Ignoring the diversity of technologies among producers can lead to incorrect conclusions regarding inefficiency measures, as demonstrated by Casu and Molyneux (Reference Casu and Molyneux2003) and Bos and Schmiedel (Reference Bos and Schmiedel2007). Brown and Glennon (Reference Brown and Glennon2000) noted that assuming a uniform production technology for all producers is a highly restrictive assumption. According to Tsionas (Reference Tsionas2002), assuming a common technology for all producers might result in a producer being ranked as inefficient, even though the producer employs a different technology and fully utilizes its own resources. Mester (Reference Mester1997), Greene (Reference Greene and Greene1993, Reference Greene2005b) and Caiazza et al. (Reference Caiazza, Pozzolo and Trovato2016) confirmed that heterogeneity leads to biased estimates obtained from the stochastic frontier approach.

Figure 7. Heterogeneous technologies and inefficiency.

To illustrate the importance of accommodating heterogeneity in the production frontier when estimating inefficiency, consider an example involving two producers, labeled A and B, with production frontiers denoted as $A$ and $B$ , respectively, as shown in Figure 7. If a common frontier $C$ is assumed for these two producers, the directional measure of overall technical inefficiency is given by $TI_{T}^{pooled}(x_{A},y_{A})=\left \Vert AC^{T}\right \Vert /\left \Vert 0g\right \Vert$ for producer $A$ and $TI_{T}^{pooled}(x_{B},y_{B})=\left \Vert BC^{T}\right \Vert /\left \Vert 0g\right \Vert$ for producer $B$ . Under this assumption, producer $A$ appears less efficient than producer $B$ because $\left \Vert AC^{T}\right \Vert \gt \left \Vert BC^{T}\right \Vert$ . However, when considering each producer operating on their own frontier, the measures become $TI_{T}^{own}(x_{A},y_{A})=\left \Vert AA^{T}\right \Vert /\left \Vert 0g\right \Vert$ and $\ TI_{T}^{own}(x_{B},y_{B})=\left \Vert BB^{T}\right \Vert /\left \Vert 0g\right \Vert$ . Consequently, the efficiency ranking of these two producers reverses, as in this case, producer $B$ appears less efficient than producer $A$ because $\left \Vert BB^{T}\right \Vert \gt \left \Vert AA^{T}\right \Vert$ . Similarly, the directional measure of output technical inefficiency is expressed as $TI_{O}^{pooled}(x_{A},y_{A})=\left \Vert AC_{A}^{O}\right \Vert$ for producer $A$ and $TI_{O}^{pooled}(x_{B},y_{B})=\left \Vert BC_{B}^{O}\right \Vert$ for producer $B$ , assuming a common frontier $C$ . Under this assumption, producer $ A$ is less efficient than producer $ B$ . However, when each producer operates on their individual frontier, the measures become $TI_{O}^{own}(x_{A},y_{A})=\left \Vert AA^{O}\right \Vert$ and $TI_{O}^{own}(x_{B},y_{B})=\left \Vert BB^{O}\right \Vert$ . Consequently, the efficiency ranking of these two producers is reversed. In this situation, producer $B$ is less efficient than producer $A$ because $\left \Vert BB^{O}\right \Vert \gt \left \Vert AA^{O}\right \Vert$ .

The directional measure of input technical inefficiency is expressed as $TI_{I}^{pooled}(x_{A},y_{A})=\left \Vert AC_{A}^{I}\right \Vert$ for producer $A$ and $TI_{I}^{pooled}(x_{B},y_{B})=\left \Vert BC_{B}^{I}\right \Vert$ for producer $B$ , assuming a common frontier $C$ . Under this assumption, producer $A$ is less efficient than producer $B$ . However, when each producer operates on their individual frontier, the measures become $TI_{I}^{own}(x_{A},y_{A})=\left \Vert AA^{I}\right \Vert$ and $TI_{I}^{own}(x_{B},y_{B})=\left \Vert BB^{I}\right \Vert$ . Consequently, the efficiency ranking of these two producers is reversed. In this situation, producer $B$ is less efficient than producer $A$ because $\left \Vert BB^{I}\right \Vert \gt \left \Vert AA^{I}\right \Vert$ .

There are several approaches to addressing heterogeneous technologies. One approach involves introducing a producer-specific intercept into the model, as demonstrated by Greene (Reference Greene2005a, Reference Greene2005b). Cornwell et al. (Reference Cornwell, Schmidt and Sickles1990) and Swamy and Tavlas (Reference Swamy and Tavlas1995) assumed that both the intercept and the slope parameters are random. Akhavein et al. (Reference Akhavein, Swamy and Taubman1997), Tsionas (Reference Tsionas2002), and Feng et al. (Reference Feng, Wang and Serletis2018) proposed a stochastic frontier model with random coefficients. Alternatively, varying coefficient models, in which the coefficients are expressed as functions of other variables, can be employed, as shown by Hastie and Tibshirani (Reference Hastie and Tibshirani1993) and Tran (Reference Tran2014).

Another approach involves categorizing producers into groups based on factors such as size, ownership, organizational structure, or geographic regions, and then estimating a model for each group. This method is illustrated by Mester (Reference Mester1996) and Altunbas et al. (Reference Altunbas, Evans and Molyneux2001), among others. However, a limitation of this approach is that the model for each group is estimated independently, without incorporating information from producers in other groups, as noted by Greene (Reference Greene and Greene1993, Reference Greene2004b), Orea and Kumbhakar (Reference Orea and Kumbhakar2004), and Parmeter and Kumbhakar (Reference Parmeter and Kumbhakar2014). Alternatively, threshold models categorize producers into technology groups based on the value of the threshold variable. The model for each group is estimated using information provided by producers in other groups. For examples of a single threshold model, see Hansen (Reference Hansen1999, Reference Hansen2000) and Yélou et al. (Reference Yélou, Larue and Tran2010); for multiple threshold models, refer to Almanidis (Reference Almanidis2013) and Tsionas et al. (Reference Tsionas, Tran and Michaelides2019). However, Almanidis (Reference Almanidis2013) noted that the joint estimation of the threshold parameters requires a grid search over an enormous number of points, which increases with the number of break points. The solution is to use sequential estimation of the threshold parameters. This method, however, yields asymptotically efficient estimates only for the last threshold parameter in the process. Bai (Reference Bai1997) introduced a refinement for estimating threshold parameters. This method involves re-estimating the threshold parameters in reverse order while keeping the estimates of the previous thresholds constant. The refined estimator has been demonstrated to be asymptotically efficient.

5.2.2 Heterogeneity in the inefficiency term

Ignoring the presence of heterogeneity in the inefficiency term can result in inaccurate measures of inefficiency, as heterogeneity not accounted for by producer-specific characteristics is incorrectly attributed to inefficiency. This heterogeneity can be addressed by incorporating producer-specific characteristics into the mean, variance, or both parameters of the inefficiency distribution. For more details on these models, refer to Section 3.

To summarize, exogenous factors that influence a producer’s output but are not considered inefficiencies, because they are beyond the producer’s control, are intended to capture technological differences and diversity. These factors should be specified in the production frontier itself. When exogenous factors that can be managed by the producer are more related to inefficiency, inefficiency heterogeneity directly impacts inefficiency and is often included in the location or scale parameters of the inefficiency distribution.