2.1. The Directional Distance Functions

The directional technology distance function (DTDF) allows for simultaneous contraction of inputs and expansion of outputs in terms of a direction vector $g$ $=\left ( g_{x},g_{y}\right )$ , where $g_{x}\in R_{+}^{N}$ and $g_{y}\in R_{+}^{M}$ such that it contracts inputs in the direction $g_{x}$ and expands outputs in the direction $g_{y}$ . In particular, the DTDF is defined as:

(1) \begin{align} \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 \} \nonumber \\[3pt] & =\max _{\theta _{T}}\!\left \{ \theta _{T}\,{:}\,\!\left ( x,y\right ) +\theta _{T}\!\left ( g_{x},g_{y}\right ) \in T\right \}. \end{align}

Efficient banks who produce on the frontier of $T$ have $\vec{D}_{T}\!\left ( x,y;\ g_{x},g_{y}\right ) =0$ . Inefficiency is indicated by $\vec{D}_{T}\!\left ( x,y;\ g_{x},g_{y}\right ) \gt 0$ , with higher values indicating greater inefficiency when banks operate beneath the frontier of $T$ . A technology-oriented measure of technical inefficiency is defined as $TI_{T}=\vec{D}_{T}\!\left ( x,y;\ g_{x},g_{y}\right )$ .

As noted by Chambers et al. (Reference Chambers, Chung and Färe1998), the DTDF is nonnegative, nondecreasing in $x$ , nonincreasing in $y$ , and concave in $(x,y)$ . Moreover, it satisfies the following translation property:

(2) \begin{equation} \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, \end{equation}

where $\alpha \in R$ is an arbitrary scaling factor. By setting $g_{y}=0$ , the directional vector becomes $g$ $=\left ( g_{x},0\right )$ and allows only for input contraction holding outputs fixed—see Figure 1 and 2. In this case, equation (1) becomes the directional input distance function (DIDF), $\vec{D}_{T}\!\left ( x,y;\ g_{x},0\right ) =\vec{D}_{I}\!\left ( y,x;\ g_{x}\right )$ . The DIDF serves as an input-oriented measure of technical inefficiency; $TI_{I}=\vec{D}_{I}\!\left ( y,x;\ g_{x}\right )$ :

Figure 1. Inefficiency measures with the observed input−output directional vector.

Figure 2. Inefficiency measures with the unit value directional vector.

(3) \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(y)\right \} =\max _{\theta _{I}}\!\left \{ \theta _{I}\,{:}\,\!\left ( x-\theta _{I}g_{x}\text{, }y\right ) \in T\right \}. \end{equation}

By setting $g_{x}=0$ , the directional vector becomes $g$ $=\left ( 0,g_{y}\right )$ and allows only for output expansion holding inputs fixed—see Figures 1 and 2. In this case, equation (1) becomes the directional output distance function (DODF), $\vec{D}_{T}\!\left ( x,y;\ 0,g_{y}\right ) =\vec{D}_{O}\!\left ( x,y;\ g_{y}\right )$ . The DODF serves as an output-oriented measure of technical inefficiency; $TI_{O}=\vec{D}_{O}\!\left ( x,y;\ g_{y}\right )$ :

(4) \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(x)\right \} =\max _{\theta _{o}}\!\left \{ \theta _{O}\,{:}\,\!\left ( x,y+\theta _{O}g_{y}\right ) \in T\right \}. \end{equation}

2.2. Duality Relationships

The directional distance functions are primal representations of the technology. The dual representations of the technology are given by the profit $\left ( \pi \!\left ( p,w\right ) \right )$ , cost $\left ( C\left ( y,w\right ) \right )$ , and revenue $\left ( R\left ( x,p\right ) \right )$ functions. The relationship between the DTDF (DIDF or DODF) and the profit (cost or revenue) functions can be represented as [see Chambers et al. (Reference Chambers, Chung and Färe1998)]:

\begin{align*} \vec {D}_{T}\!\left ( x,y;\ g_{x},g_{y}\right ) & \leq \frac {\pi\!\left ( p,w\right ) -\left ( py\text { }-\text { }wx\right ) }{pg_{y}+wg_{x}};\ \vec {D}_{I}\!\left ( y,x;\ g_{x}\right )\\[3pt] &\leq \frac {wx-C\left ( y,w\right ) }{wg_{x}};\ \vec {D}_{O}\left ( x,y\text {;}g_{y}\right ) \leq \frac {R\!\left ( x,p\right ) -py}{pg_{y}}.\end{align*}

The right-hand side can be interpreted as a measure of profit (cost or revenue) inefficiency comparing observed profit (cost or revenue) to maximum profit (minimum cost or maximum revenue) normalized by the value of the directional vector. The left-hand side captures overall (input or output) technical inefficiency, respectively. The inequality can be turned into equality by adding a residual term that captures allocative inefficiency, where allocative inefficiency is due to the failure of choosing the profit-maximizing (cost-minimizing or revenue-maximizing) input−output (input or output) vector given relative input and output market prices. Thus, technical inefficiency is due to the overuse of inputs or the loss of production of outputs or both, and allocative inefficiency results from employing inputs and outputs in the wrong proportions.

2.3. Exogenous Directional Vectors

Two widely used exogenous directions are the observed input−output direction and the unit value direction. The observed input−output direction $g$ $=\left ( -x,y\right )$ assumes that an inefficient bank can decrease inefficiency while decreasing input and increasing output in proportion to the initial combination of the actual input and output. Figure 1 illustrates how inefficiency can be measured using this type of directional vector. Banks who operate at point $A$ are technically inefficient. The simultaneous maximum proportional contraction of input and expansion of output can be measured in terms of the lengths of $x$ and $y$ , with the use of the Pythagorean theorem, as:

\begin{equation*} \vec {D}_{T}\left ( x,y;\ g_{x},g_{y}\right ) =\vec {D}_{T}\left ( x,y;\ -x,y\right ) =\frac {\left \Vert AB\right \Vert }{\left \Vert 0g\right \Vert }=\frac {\sqrt {\left ( \left \Vert x\right \Vert -\left \Vert x^{T}\right \Vert \right ) ^{2}+\left ( \left \Vert y^{T}\right \Vert -\left \Vert y\right \Vert \right ) ^{2}}}{\sqrt {\left \Vert x\right \Vert ^{2}+\left \Vert y\right \Vert ^{2}}}=\theta _{T}.\end{equation*}

The maximum proportional contraction of input holding output constant can be measured using the directional vector $g$ $=\left ( -x,0\right )$ . Technical inefficiency is considered to be input-oriented technical inefficiency and is defined by the difference of the lengths of $x$ and $x^{I}$ divided by the length of $x$ :

\begin{equation*} \vec {D}_{T}\left ( x,y;\ -x,0\right ) =\vec {D}_{I}\left ( y,x;\ -x\right ) =\frac {\left \Vert x\right \Vert -\left \Vert x^{I}\right \Vert }{\left \Vert x\right \Vert }=1-\frac {\left \Vert x^{I}\right \Vert }{\left \Vert x\right \Vert }=1-\frac {1}{D_{I}\left ( y,x\right ) }=\theta _{I},\end{equation*}

where $D_{I}\left ( y,x\right )$ is the standard input distance function. The maximum proportional expansion of output holding input constant can be measured using the directional vector $g$ $=\left ( 0,y\right )$ . In this case, technical inefficiency is considered to be output-oriented technical inefficiency and is defined as:

\begin{equation*} \vec {D}_{T}\left ( x,y;\ 0,y\right ) =\vec {D}_{O}\left ( x,y;\ y\right ) =\frac {\left \Vert y^{O}\right \Vert -\left \Vert y\right \Vert }{\left \Vert y\right \Vert }=\frac {\left \Vert y^{O}\right \Vert }{\left \Vert y\right \Vert }-1=\frac {1}{D_{O}\left ( x,y\right ) }-1=\theta _{O},\end{equation*}

where $D_{O}\left ( x,y\right )$ is the standard output distance function. A critical question that needs to be considered is whether $\theta _{T}=\theta _{I}+\theta _{O}?$

$\theta _{I}$ arises from employing the wrong level of inputs and measures the maximum contraction of a bank’s input to that of a best-practice or most efficient bank producing the same output. $\theta _{O}$ arises from producing at the wrong level of outputs and measures the maximum expansion of a bank’s output to that of a best-practice or most efficient bank using the same input. The interactions between input and output inefficiencies, $\theta _{IO}$ , captures output (input) implications of any errors in the input usage (output production).

Another widely used prespecified direction is the unit value direction $g$ $=\left ( -1,1\right )$ . This type of directional vector implies that the amount by which a bank could decrease input and increase output will be $\vec{D}_{T}\!\left ( x,y;\ -1,1\right ) \times 1$ units of $x$ and $y$ . Figure 2 illustrates how inefficiency can be measured using the unit value direction. Banks who operate at point $A$ are technically inefficient. The simultaneous maximum contraction of input and expansion of output can be measured as:

\begin{equation*} \vec {D}_{T}\left ( x,y;\ -1,1\right ) =\frac {\left \Vert AB\right \Vert }{\left \Vert 0g\right \Vert }=\frac {\sqrt {\left ( \left \Vert x\right \Vert -\left \Vert x^{T}\right \Vert \right ) ^{2}+\left ( \left \Vert y^{T}\right \Vert -\left \Vert y\right \Vert \right ) ^{2}}}{\sqrt {2}}=\theta _{T}^{1}.\end{equation*}

The maximum contraction of input holding output constant can be measured using the directional vector $g$ $=\left ( -1,0\right )$ . Technical inefficiency is considered to be input-oriented technical inefficiency and is defined as $\vec{D}_{I}\!\left ( y,x;\ -1\right ) =\left \Vert AI\right \Vert =\left \Vert x\right \Vert -\left \Vert x^{I}\right \Vert =\theta _{I}^{1}$ .

The maximum expansion of output holding input constant can be measured using the directional vector $g$ $=\left ( 0,1\right )$ . In this case, technical inefficiency is considered to be output-oriented technical inefficiency and is defined as $\vec{D}_{O}\!\left ( x,y;\ 1\right ) =\left \Vert AO\right \Vert =\left \Vert y^{O}\right \Vert -\left \Vert y\right \Vert =\theta _{O}^{1}$ . Does $\theta _{T}^{1}=\theta _{I}^{1}+\theta _{O}^{1}?$

Corollary 1 Let $\theta _{T}^{1}$ be overall technical inefficiency, $\theta _{I}^{1}$ be input-oriented technical inefficiency, and $\theta _{O}^{1}$ be output-oriented technical inefficiency derived using the unit value directional vectors. Then the IEIO, $\theta _{IO}^{1}$ , is related to $\theta _{I}^{1}$ and $\theta _{O}^{1}$ as:

\begin{align*} \theta _{IO}^{1} & =\theta _{I}^{1}+\left ( \left \Vert y^{O}\right \Vert -\left \Vert y^{T}\right \Vert \right ),\\[3pt] \theta _{IO}^{1}&=\theta _{O}^{1}+\left ( \left \Vert x^{T}\right \Vert -\left \Vert x^{I}\right \Vert \right ).\end{align*}

While banks who operate on the production frontier have zero IEIO since $\theta _{T}^{1}=\theta _{I}^{1}=\theta _{O}^{1}\,{=}\,0$ , banks who operate beneath the production frontier have negative IEIO since $\theta _{T}^{1}\lt \theta _{I}^{1}+\theta _{O}^{1}$ . Furthermore, there is a relationship between the IEIO, $\theta _{IO}^{1}$ , and the input and output technical inefficiencies. The IEIO equals the input technical inefficiency $\theta _{I}^{1}$ plus the loss of production of output $\left \Vert y^{O}\right \Vert -\left \Vert y^{T}\right \Vert$ (i.e., part of output technical inefficiency) that is forgone to reduce input and eliminate part of the input technical inefficiency $\left \Vert x\right \Vert -\left \Vert x^{T}\right \Vert$ —see Figure 2. This suggests that the loss of output creates output technical inefficiency and has an effect on reducing input technical inefficiency. That is, a loss of bank output creates a loss of revenue and has an effect on the reduction of the use of inputs. The IEIO also equals the output technical inefficiency $\theta _{O}^{1}$ plus the overuse of input $\left \Vert x^{T}\right \Vert -\left \Vert x^{I}\right \Vert$ (i.e., part of input technical inefficiency) that is used to produce more output and eliminate part of the output technical inefficiency $\left \Vert y^{T}\right \Vert -\left \Vert y\right \Vert$ —see Figure 2. This suggests that the overuse of input creates input technical inefficiency and has an effect on reducing output technical inefficiency. That is, an overuse of a certain input creates additional resources that can be used to produce more outputs that are intensive in using that input.

2.4. Endogenous Directional Vectors

When information on input and output prices is available and banks are assumed to exhibit cost-minimizing (or revenue or profit-maximizing) behavior, technical inefficiency can be measured by choosing an endogenous directional vector such that it projects any inefficient bank to the cost-minimizing (or revenue or profit-maximizing) benchmark.

Following Zofio et al. (Reference Zofio, Pastor and Aparicio2013), the directional vector $g$ $=\left ( g_{x}^{\pi },g_{y}^{\pi }\right )$ is assumed to satisfy the price normalization constraint $pg_{y}^{\pi }+wg_{x}^{\pi }=1$ and projects any inefficient bank towards the profit-maximizing bundle $\left ( x^{\pi },y^{\pi }\right )$ where banks are both technically and allocatively efficientFootnote ¹ . Thus, the directional vector can be defined as:

(5) \begin{equation} g=\left ( g_{x}^{\pi },g_{y}^{\pi }\right ) =\left ( \frac{x-x^{\pi }}{\pi \!\left ( p,w\right ) -\left ( py-wx\right ) }\text{, }\frac{y^{\pi }-y}{\pi \!\left ( p,w\right ) -\left ( py-wx\right ) }\right ), \end{equation}

to ensure that $pg_{y}^{\pi }+wg_{x}^{\pi }=1$ . Then, the DTDF, $\vec{D}_{T}\!\left ( x,y;\ g_{x}^{\pi },g_{y}^{\pi }\right )$ , equals the loss of profit due to technical inefficiency and gives a measure of overall technical inefficiency in monetary values.

Similar cost (revenue) results can be obtained by choosing endogenous directional vectors that satisfy the price normalization constraint and projects any inefficient bank towards the cost-minimizing (revenue-maximizing) bundle.

Does $\theta _{T}^{\pi }=\theta _{I}^{C}+\theta _{O}^{R}?$

Consequently, including price information on the directional vector such as the directional vector that projects any inefficient bank to the profit-maximizing bundle, the IEIO may have positive or negative effects on the overall technical inefficiency, depending on the relationship between cost and profit inefficiency. Cost and profit inefficiency may be positively related if bank managers aim for both low cost and profit inefficiency. Cost and profit inefficiency may be negatively related if high-quality services require higher costs and result in higher measured cost inefficiency but get higher output prices that result in higher profits and lower measured profit inefficiency (see Rogers (Reference Rogers1998) and Maudos et al. (Reference Maudos, Pastor, Perez and Quesada2002)).

3.1. Model Specification

To obtain the estimates of the directional distance functions and therefore the measure of technical inefficiencies, this subsection provides the parametric specification of these functions. This involves choosing a functional form, imposing the parameter restrictions for the translation property, modeling the interactive effect, and specifying the directional vector $g$ .

3.1.2. Imposing the restrictions

The translation property can be imposed by setting $\alpha$ equal to an arbitrarily chosen input or the negative of an arbitrarily chosen output which is specific to each bank, say $\alpha =-y_{1}$ , and normalizing the corresponding directional vector $g_{y_{1}}=1$ —see, for example, Malikov et al. (Reference Malikov, Kumbhakar and Tsionas2016)Footnote ² . Using this transformation process and applying it to the empirical implementation that uses two inputs to produce two outputs, the translation property in equation (2) can be rewritten as:

(6) \begin{equation} \vec{D}_{T}\!\left ( x+y_{1}g_{x},y_{2}-y_{1}g_{y_{2}},t;\ g_{x},g_{y}\right ) =\vec{D}_{T}\!\left ( x,y,t;\ g_{x},g_{y}\right ) +y_{1}. \end{equation}

Note that the output $y_{1}$ disappears from the left-hand side of (6) because of $y_{1}-y_{1}(1)=0$ . Rearranging and adding a random error $v_{T}$ to equation (6) yields the standard stochastic frontier model with two error terms, as follows:

(7) \begin{equation} y_{1}=\vec{D}_{T}\!\left ( x+y_{1}g_{x},y_{2}-y_{1}g_{y_{2}},t;\ g_{x},g_{y}\right ) +v_{T}-u_{T}, \end{equation}

where $v_{T}$ is a two-sided random error assumed to be identically and independently distributed (iid) with mean zero and variance $\sigma _{v_{T}}^{2}=\Sigma,v_{T}\sim N\left ( 0,\Sigma \right )$ and $\vec{D}_{T}\!\left ( x,y,t;\ g_{x},g_{y}\right ) =u_{T}\geq 0$ is a one-sided error term which captures bank-specific overall technical inefficiency. Applying the quadratic functional form to the first term on the right-hand side of (7), (7) can be written as:

(8) \begin{align} y_{1} & =\alpha _{0}+\sum _{n=1}^{2}\alpha _{n}\widetilde{x}_{n}+\beta _{2}\widetilde{y}_{2}+\delta _{t}t+\frac{1}{2}\sum _{n=1}^{2}\sum _{n^{\prime }=1}^{2}\alpha _{nn^{\prime }}\widetilde{x}_{n}\widetilde{x}_{n^{\prime }}+\frac{1}{2}\beta _{22}\left (\,\widetilde{y}_{2}\right ) ^{2}\nonumber \\[3pt] & \quad + \frac{1}{2}\delta _{tt}t^{2}+\sum _{n=1}^{2}\gamma _{n2}\widetilde{x}_{n}\widetilde{y}_{2}+\sum _{n=1}^{2}\delta _{tx_{n}}t\widetilde{x}_{n}+\delta _{ty_{2}}t\widetilde{y}_{2}+v_{T}-u_{T}, \end{align}

where $\widetilde{x}_{n}=x_{n}+y_{1}g_{x_{n}}$ $\left ( n=1,2\right )$ , $\widetilde{y}_{2}=y_{2}-y_{1}g_{y_{2}}$ , and $y_{1}$ corresponds to the dependent variable. The parameters of (8) must satisfy the usual restrictions for symmetry $\alpha _{nn^{\prime }}=\alpha _{n^{\prime }n}$ $(n\neq n^{\prime })$ . Within a panel data framework, the DTDF model in (8) can be notationally simplified as:

(9) \begin{equation} y_{1},_{it}=\widetilde{R}_{T,it}(g)^{\prime }\beta _{T}+v_{T,it}-u_{T,it}, \end{equation}

where $i=1,\ldots,K$ indicates banks; $t=1,\ldots,T$ indicates time; $\widetilde{R}_{T}(g)$ , which is a function of $g$ , is a vector of all the relevant variables on the right-hand side of (8) including a unity for the intercept term; and $\beta _{T}$ is the corresponding vector of coefficients (including the intercept).

Similarly, the translation property of the DIDF can be imposed by setting $\alpha$ equal to an arbitrarily chosen input which is specific to each bank, say $\alpha =x_{1}$ , and normalizing the corresponding directional vector $g_{x_{1}}=1$ . Using this transformation process and following the above methodology, the quadratic functional form can be written as:

(10) \begin{align} -x_{1} & =\alpha _{0}+\alpha _{2}\widetilde{x}_{2}+\sum _{m=1}^{2}\beta _{m}y_{m}+\delta _{t}t+\frac{1}{2}\alpha _{22}\left (\,\widetilde{x}_{2}\right ) ^{2}+\frac{1}{2}\sum _{m=1}^{2}\sum _{m^{\prime }=1}^{2}\beta _{mm^{\prime }}y_{m}y_{m^{\prime}}\nonumber \\[3pt] & \quad +\frac{1}{2}\delta _{tt}t^{2}+\sum _{m=1}^{2}\gamma _{2m}\widetilde{x}_{2}y_{m}+\delta _{tx_{2}}t\widetilde{x}_{2}+\sum _{m=1}^{2}\delta _{ty_{m}}ty_{m}+v_{I}-u_{I}, \end{align}

where $\widetilde{x}_{2}=x_{2}-x_{1}g_{x_{2}}$ , $x_{1}$ corresponds to the dependent variable, $u_{I}\geq 0$ is a one-sided error term which captures bank-specific input technical inefficiency. The parameters of (10) must satisfy the usual restrictions for symmetry $\beta _{mm^{\prime }}=\beta _{m^{\prime }m}$ $(m\neq m^{\prime })$ . Within a panel data framework, the DIDF model in (10) can be notationally simplified as:

(11) \begin{equation} -x_{1},_{it}=\widetilde{R}_{I,it}(g)^{\prime }\beta _{I}+v_{I,it}-u_{I,it} \end{equation}

When $g_{x}=0$ , the DTDF model in (8) reduces to the DODF that allows for only output expansion:

(12) \begin{align} y_{1} & =\alpha _{0}+\sum _{n=1}^{2}\alpha _{n}x_{n}+\beta _{2}\widetilde{y}_{2}+\delta _{t}t+\frac{1}{2}\sum _{n=1}^{2}\sum _{n^{\prime }=1}^{2}\alpha _{nn^{\prime }}x_{n}x_{n^{\prime }}+\frac{1}{2}\beta _{22}\left (\,\widetilde{y}_{2}\right ) ^{2}\nonumber \\[3pt] &\quad +\frac{1}{2}\delta _{tt}t^{2}+\sum _{n=1}^{2}\gamma _{n2}x_{n}\widetilde{y}_{2}+\sum _{n=1}^{2}\delta _{tx_{n}}tx_{n}+\delta _{ty_{2}}t\,\widetilde{y}_{2}+v_{O}-u_{O}, \end{align}

where $u_{O}\geq 0$ is a one-sided error term which captures output technical inefficiency. Within a panel data framework, the DODF model in (12) can be notationally simplified as:

(13) \begin{equation} y_{1},_{it}=\widetilde{R}_{O,it}(g)^{\prime }\beta _{O}+v_{O,it}-u_{O,it}. \end{equation}

3.1.3. Modeling the interactive effect

Following Battese and Coelli (Reference Battese and Coelli1995), overall technical inefficiency, $u_{T,it}$ , can be modeled as a linear function of a vector of explanatory bank-specific variables $Z_{it}$ that are expected to influence $u_{T,it}$ . Bank-specific variables $Z_{it}$ include input technical inefficiency, $u_{I,it}$ , output technical inefficiency, $u_{O,it}$ , and a term capturing the interactions between them, $u_{I,it}\times u_{O,it}$ :

(14) \begin{equation} u_{T,it}=Z_{it}\delta +v_{u,it}, \end{equation}

where $\delta$ is an unknown vector of coefficients (including the intercept) to be estimated, and $v_{u,it}$ is an error term that is defined by the truncation of a normal distribution. It is possible to allow the explanatory variables $Z_{it}$ to affect the production frontier by adding interaction terms between $Z_{it}$ and the regressors, $Z_{it}X_{it}$ , similar to Huang and Liu (Reference Huang and Liu1994). However, this is not the focus of this paper.

3.1.4. Specifying the directional vector

As discussed earlier in Section 2, there are two approaches in the literature. The first is to choose the directional vector a priori. The second approach is to let the data determine the directional vector. In this paper, both approaches are used and compared.

The prespecified directional vector

Two widely used prespecified directions are the unit value direction—see, for example, Park and Weber (Reference Park and Weber2006) and Koutsomanoli-Filippaki et al. (Reference Koutsomanoli-Filippaki, Margaritis and Staikouras2009), and the observed input−output direction—see, for example, Färe et al. (Reference Färe, Grosskopf and Weber2004). Input-oriented measures of technical inefficiency $u_{I,it}$ using the unit value (or the observed input) direction can be obtained by setting $g_{x}=1$ $\left ( \text{or }g_{x_{1}}=1\text{, and }g_{x_{2}}=x_{2}\right )$ in the case of (11) and estimate the single-equation DIDF subject to the usual symmetry restrictions and theoretical monotonicity restrictions. Similarly, output-oriented measures of technical inefficiency $u_{O,it}$ using the unit value (or the observed output) direction can be obtained by setting $g_{y}=1$ $\left ( \text{or }g_{y_{1}}=1\text{, and }g_{y_{2}}=y_{2}\right )$ in the case of (13) and estimate the single-equation DODF subject to the usual symmetry restrictions and theoretical monotonicity restrictions. Overall or technology-oriented measures of technical inefficiency $u_{T,it}$ using the unit value (or the observed input−output) direction can be obtained by setting $g=(g_{x},g_{y})=(-1,1)$ $\left ( \text{or }g_{x_{1}}=x_{1},g_{x_{2}}=x_{2},g_{y_{1}}=1\text{, and }g_{y_{2}}=y_{2}\right )$ in the case of (9) and estimate the system of equations that includes the DTDF and the interactive effect equation in (14) subject to the usual symmetry restrictions and theoretical monotonicity restrictions, while the translation property is already imposed by construction. More specifically, the system can be written as:

\begin{equation*} \left [ \begin {array} [c]{c}y_{1},_{it}\\[3pt] u_{T,it}\end {array} \right ] =\left [ \begin {array} [c]{cc}\widetilde {R}_{T,it}(g)^{\prime } & \\[3pt] & Z_{it}^{\prime }\end {array} \right ] \left [ \beta \right ] -\left [ \begin {array} [c]{c}u_{T,it}\\[3pt] 0 \end {array} \right ] +\left [ \begin {array} [c]{c}v_{T,it}\\[3pt] v_{u,it}\end {array} \right ] \end{equation*}

which can be written in a compressed form as:

(15) \begin{equation} Y_{it}=R_{it}(g)\beta -u_{T,it}\iota +v_{it}, \end{equation}

where $\iota =\left [ 1,0\right ]$ and $v_{it}=\left ( v_{T,it},v_{u,it}\right ) ^{\prime }\sim N\!\left ( 0,\Sigma \right )$ . Bank-specific variables $Z$ is a vector of the relevant variables on the right-hand side of (14) that are obtained using the unit value (or the observed input and output) directional vector in equations (11) and (13). In this system of equations, bank-specific variables $Z$ that determine overall technical inefficiency are estimated simultaneously with the variables that determine the frontier.

The optimal directional vector

The optimal directional vector is treated as unknown parameters in which the bank’s movement toward the efficient frontier is to be estimated.

The cost-optimal directional vector

The bank cost-minimizing objective can be defined in terms of the DIDF in equation (3), to keep consistency with the endogeneity of inputs and exogeneity of outputs, as:

(16) \begin{equation} C(y,w)=\underset{x}{\min }\left \{ w^{\prime }x\,{:}\,\vec{D}_{I}\!\left ( y,x;\ g_{x}\right ) \geq 0\right \}. \end{equation}

Following Luenberger (Reference Luenberger1992), the problem in (16) can be represented by an unconstrained problem as:

\begin{equation*} C(y,w)=\underset {x}{\min }\left \{ w^{\prime }x-\vec {D}_{I}\left ( y,x;\ g_{x}\right ) \times w^{\prime }g_{x}\right \}. \end{equation*}

The corresponding first-order conditions are:

(17) \begin{equation} w_{n}=\nabla _{x_{n}}\vec{D}_{I}(.)\lambda _{I} {\rm{for}} \text{ }n=1,2, \end{equation}

where $\lambda _{I}=w^{\prime }g_{x}=\sum _{n=1}^{N}w_{n}g_{x_{n}}$ is the sum of direction-weighted cost. The first-order conditions in (17) are the inverse demand functions. To meet the rank condition for the identification of the model, a total of at least the total number of potentially endogenous variables is needed as independent equations in the system. As it is well known, the system (17) is singular and only $(n-1)$ equations in (17) can be used for the estimation—see Barten (Reference Barten1969) for more details. The DIDF in (11) plus the $(n-1)$ first-order conditions in (17) provide a system of $n$ equations. Precisely, the system consists of the DIDF in (11) and the first-order condition for $w_{2}$ . Note that $\nabla _{x_{2}}\vec{D}_{I}\!\left ( y,\widetilde{x};\ g_{x}\right ) =\nabla _{x_{2}}\vec{D}_{I}\!\left ( y,x;\ g_{x}\right )$ by the translation property, then the first-order condition in (17) can be rewritten in terms of the parameters of (11) after adding an iid normal error term $v_{C}$ as follows:

(18) \begin{equation} w_{2}=\lambda _{I}\left ( \alpha _{2}+\alpha _{22}\widetilde{x}_{2}+\sum _{m=1}^{2}\gamma _{2m}y_{m}+\delta _{tx_{2}}t\right ) \mathbf{+}v_{C}. \end{equation}

Note that, solving the first-order condition for $\widetilde{x}_{2}$ treats $\widetilde{x}_{2}$ as an endogenous variable (as opposed to $x_{2}$ ). The equivalence of working with $\widetilde{x}_{2}$ and working with $x_{2}$ holds because of $\partial \widetilde{x}_{2}/$ $\partial x_{2}=1$ . By allowing $g$ to differ across banks, (18) can be further written in a panel data framework as:

(19) \begin{equation} w_{2,it}=\widetilde{R}_{C},_{it}(g_{i})^{\prime }\beta _{C}+v_{C,it}, \end{equation}

where $\widetilde{R}_{C}$ is a vector of the relevant variables on the right-hand side of (18) and $\beta _{C}$ is the corresponding vector of coefficients. $\beta _{C}$ is a subset of $\beta _{I}$ , which can be obtained using a selection matrix $A_{I}$ that contains elements which are either 0 or 1, where $\beta _{C}=A_{I}\beta _{I}$ .

The DIDF system (equations (11) and (19)) is a simultaneous equation model where the entire vector $x$ is endogenous. The entire system can be written as:

\begin{equation*} \left [ \begin {array} [c]{c}-x_{1},_{it}\\[3pt] w_{2,it}\end {array} \right ] =\left [ \begin {array} [c]{cc}\widetilde {R}_{I,it}(g_{i})^{\prime } & \\[3pt] & \widetilde {R}_{C,it}(g_{i})^{\prime }\end {array} \right ] \left [ \beta \right ] -\left [ \begin {array} [c]{c}u_{I,it}\\[3pt] 0 \end {array} \right ] +\left [ \begin {array} [c]{c}v_{I,it}\\[3pt] v_{C,it}\end {array} \right ] \end{equation*}

which can be written in a compressed form as:

(20) \begin{equation} Y_{it}=R_{it}(g_{i})\beta -u_{I,it}\iota +v_{it}, \end{equation}

where $\iota =\left [ 1,0\right ]$ , $v_{it}=\left ( v_{I,it},v_{C,it}\right ) ^{\prime }\sim N\left ( 0,\Sigma \right )$ , and $g_{i}=\left ( g_{x_{2i}}\right ) ^{^{\prime }}$ with $i=1,\ldots,K$ indexing banks. The system is estimated subject to the symmetry restrictions and theoretical monotonicity restrictions. The directional vector $g_{i}$ is treated as unknown parameters which are estimated jointly with the remaining parameters in the system. The obtained estimates of the directional vector can be interpreted as being cost-optimal due to the inclusion of the cost-minimizing first-order conditions in the system—see Malikov et al. (Reference Malikov, Kumbhakar and Tsionas2016). That is, the estimated DIDF direction captures the bank movement to the point on a technological frontier where costs are minimized.

The revenue-optimal directional vector

The bank revenue-maximizing objective can be defined in terms of the DODF in equation (4), to keep consistency with the endogeneity of outputs and exogeneity of inputs, as:

\begin{equation*} R\left ( x,p\right ) =\underset {y}{\max }\left \{ p^{\prime }y+\vec {D}_{O}\!\left ( x,y;\ g_{y}\right ) \times p^{\prime }g_{y}\right \} \end{equation*}

The first-order conditions for the revenue maximization problem are

(21) \begin{equation} p_{m}=-\nabla _{y_{m}}\vec{D}_{O}(.)\lambda _{O} {\rm{for}} \text{ }m=1,2 \end{equation}

where $\lambda _{O}=p^{\prime }g_{y}=\sum _{m=1}^{M}p_{m}g_{y_{m}}$ is the sum of direction-weighted revenues. The first-order conditions in (21) are the inverse supply functions. A system of $m$ equations consists of the DODF in equation (13) plus the first-order condition for $p_{2}$ in (21), which can be rewritten in terms of the parameters of (13) after adding an iid normal error term $v_{R}$ as follows:

(22) \begin{equation} p_{2}=-\lambda _{O}\left ( \beta _{2}+\beta _{22}\widetilde{y}_{2}+\sum _{n=1}^{2}\gamma _{n2}x_{n}+\delta _{ty_{2}}t\right ) \mathbf{+}v_{R}. \end{equation}

By allowing $g$ to differ across banks, (22) can be further written in a panel data framework as:

(23) \begin{equation} p_{2},_{it}=\widetilde{R}_{R},_{it}(g_{i})^{\prime }\beta _{R}+v_{R,it}, \end{equation}

where $\widetilde{R}_{R}$ is a vector of the relevant variables on the right-hand side of (22) and $\beta _{R}$ , which is a subset of $\beta _{O}$ , is the corresponding vector of coefficients.

The DODF system (equations (13) and (23)) is a simultaneous equation model where the entire vector $y$ is endogenous. The entire system can be written as:

\begin{equation*} \left [ \begin {array} [c]{c}y_{1},_{it}\\[3pt] p_{2},_{it}\end {array} \right ] =\left [ \begin {array} [c]{cc}\widetilde {R}_{O,it}(g_{i})^{\prime } & \\[3pt] & \widetilde {R}_{R,it}(g_{i})^{\prime }\end {array} \right ] \left [ \beta \right ] -\left [ \begin {array} [c]{c}u_{O,it}\\[3pt] 0 \end {array} \right ] +\left [ \begin {array} [c]{c}v_{O,it}\\[3pt] v_{R,it}\end {array} \right ] \end{equation*}

which can be written in a compressed form as:

(24) \begin{equation} Y_{it}=R_{it}(g_{i})\beta -u_{O,\ it}\iota +v_{it}, \end{equation}

where $\iota =\left [ 1,0\right ]$ , $v_{it}=\left ( v_{O,it},v_{R,it}\right ) ^{\prime }\sim N\left ( 0,\Sigma \right )$ , and $g_{i}=\left ( g_{y_{2i}}\right ) ^{^{\prime }}$ for $i=1,\ldots,K$ . The obtained estimates of the directional vector $g_{i}$ can be interpreted as being revenue-optimal due to the inclusion of the revenue-maximizing first-order conditions in the system.

The profit-optimal directional vector

Following Chambers et al. (Reference Chambers, Chung and Färe1998), the bank profit-maximizing objective can be defined in terms of the DTDF in equation (1) as:

\begin{equation*} \pi \!\left ( p,w\right ) =\max _{x,y}\left \{ \left ( p^{\prime }y-w^{\prime }x\right ) +\vec {D}_{T}\left ( x,y;\ g_{x},g_{y}\right ) (p^{\prime }g_{y}+w^{\prime }g_{x})\right \}. \end{equation*}

The first-order conditions for the profit maximization problem are

(25) \begin{align} w_{n}&=\nabla _{x_{n}}\vec{D}_{T}(.)\lambda _{T}\text{ for }n=1,\ldots,N, \nonumber\\[4pt] p_{m} &=-\nabla _{y_{m}}\vec{D}_{T}(.)\lambda _{T}\text{ for }m=1,\ldots,M, \end{align}

where $\lambda _{T}=p^{\prime }g_{y}+w^{\prime }g_{x}=\sum _{m=1}^{M}p_{m}g_{y_{m}}+\sum _{n=1}^{N}w_{n}g_{x_{n}}$ is the sum of direction-weighted profits. The first-order conditions in (25) can be rewritten in terms of the parameters of (9) after adding iid normal error terms $v_{\pi }$ as follows:

(26) \begin{align} w_{n}&=\lambda _{T}\left ( \alpha _{n}+\sum _{n^{\prime }=1}^{2}\alpha _{nn^{\prime }}\widetilde {x}_{n^{\prime }}+\gamma _{n2}\widetilde {y}_{2}+\delta _{tx_{n}}t\right ) \mathbf {+}v_{\pi w_{n}}\text { for }\left ( n=1,2\right ) \nonumber\\[3pt]p_{2}&=-\lambda _{T}\left ( \beta _{2}+\beta _{22}\widetilde{y}_{2}+\sum _{n=1}^{2}\gamma _{n2}\widetilde{x}_{n}+\delta _{ty_{2}}t\right ) \mathbf{+}v_{\pi p_{2}}.\text{ } \end{align}

By allowing $g$ to differ across banks, (26) can be further written in a panel data framework as:

(27) \begin{align} w_{n},_{it}&=\widetilde {R}_{w_{n},it}(g_{i})^{\prime }\beta _{w_{n}}+v_{\pi w_{n},it}\text { for }\left ( n=1,2\right ) \nonumber\\[3pt]p_{2},_{it}&=\widetilde{R}_{p_{2},it}(g_{i})^{\prime }\beta _{p_{2}}+v_{\pi p_{2},it},\text{ } \end{align}

where $\widetilde{R}_{w_{n}}$ and $\widetilde{R}_{p_{2}}$ are the vectors of the relevant variables on the right-hand sides of (26) and $\beta _{w_{n}}$ and $\beta _{p_{2}}$ , which are subsets of $\beta _{T}$ , are the corresponding vectors of coefficients.

The DTDF in equation (9) plus the $(\left ( n+m\right ) -1)$ first-order conditions for profit maximization given in equation (27) provide a system of $\left ( n+m\right )$ equations. This system of equations is a simultaneous equation model where the entire vector $(x,y)$ is endogenous. Adding the interactive effect equation in (14), the entire system can be written as:

\begin{equation*} \left [ \begin {array} [c]{c}y_{1},_{it}\\ w_{1},_{it}\\ w_{2},_{it}\\ p_{2},_{it}\\ u_{T,it}\end {array} \right ] =\left [ \begin {array} [c]{ccccc}\widetilde {R}_{T,it}(g_{i})^{\prime } & & & & \\ & \widetilde {R}_{w_{1},it}(g_{i})^{\prime } & & & \\ & & \widetilde {R}_{w_{2},it}(g_{i})^{\prime } & & \\ & & & \widetilde {R}_{p_{2},it}(g_{i})^{\prime } & \\ & & & & Z_{it}^{\prime }\end {array} \right ] \left [ \beta \right ] -\left [ \begin {array} [c]{c}u_{T,it}\\ 0\\ 0\\ 0\\ 0 \end {array} \right ] +\left [ \begin {array} [c]{c}v_{T,it}\\ v_{\pi w_{1},it}\\ v_{\pi w_{2},it}\\ v_{\pi p_{2},it}\\ v_{u,it}\end {array} \right ] \end{equation*}

which can be written in a compressed form as:

(28) \begin{equation} Y_{it}=R_{it}(g_{i})\beta -u_{T,it}\iota +v_{it}, \end{equation}

where $\iota\! =\!\left [ 1,0,0,0,0\right ]$ , $v_{it}\!=\!\left ( v_{T,it},v_{\pi w_{1},it},v_{\pi w_{2},it},v_{\pi p_{2},it},v_{u,it}\right ) ^{\prime }\sim N\!\left ( 0,\Sigma \right )$ , and $g_{i}\!=\!\left ( g_{x_{1i}},g_{x_{2}i},g_{y_{2i}}\right )^{\prime}$ for $i=1,\ldots,K$ . Bank-specific variables $Z$ is a vector of the relevant variables on the right-hand side of (14) that are obtained using the cost-optimal and revenue-optimal directional vectors defined in the system of equations in (20) and (24), respectively. The obtained estimates of the directional vector $g_{i}$ can then be interpreted as being profit-optimal due to the inclusion of the profit-maximizing first-order conditions in the system.

Setting all the elements of $g$ equal to ones; $g_{x_{n1}}=g_{x_{n2}}=\ldots =g_{x_{nk}}=1$ for $\left ( n=1,2\right )$ , and $g_{y_{21}}=g_{y_{22}}=\ldots =$ $g_{y_{2k}}=1$ $\left ( \text{or }g_{x_{1i}}=x_{1_{i}},g_{x_{2i}}=x_{2_{i}},\text{and }g_{y_{2_{i}}}=y_{2_{i}}\right)$ without additional first-order condition equations, the system in (28) reduces to (15) in the case of the unit value (or the observed input−output) directional vector.

4.1. Prior Distributions

The use of Bayesian approach requires choosing prior distributions for the parameters $\beta$ , $\Sigma ^{-1}$ , $u_{it}$ , $\lambda ^{-1}$ , and $g_{i}$ . For the ease of the comparison of the results among the three directional distance function models, the same prior distributions for the parameters are used. Following Gelfand et al. (Reference Gelfand, Hills, A.Racine-Poon and Smith1990), a normal prior distribution with zero mean and a large variance for $\beta$ is used to ensure that the prior distribution for $\beta$ is relatively uninformative:

(29) \begin{equation} p(\beta )\sim N(\beta _{0},\Omega _{\beta })I\left ( \beta \in S_{j}\left ( g_{i}\right ) \right ), \end{equation}

where $\beta _{0}$ is a vector of zeros and $\Omega _{\beta }$ is a diagonal matrix with $10^{4}$ in diagonal elements. $I\left ( \beta \in S_{j}\left ( g_{i}\right ) \right )$ is an indicator function which takes the value 1 if the constraints are satisfied and 0 otherwise, and $S_{j}\left ( g_{i}\right )$ , which depends on $g_{i}$ , is the set of permissible parameter values when no theoretical regularity constraints $\left ( j=0\right )$ are imposed and when the theoretical regularity constraints $\left ( j=1\right )$ must be satisfied. The indicator function restricts prior support to the region where the theoretical regularity constraints are satisfied.

For the covariance matrix $\Sigma$ , the Wishart distribution is used first due to its conjugacy properties with the normal sampling model. However, it is found to be biased toward large values which result in large values for $\Sigma$ and consequently large values for the inefficiency measures. The MCMC algorithm for a system of equations is also terminated after a small number of iterations due to the large values involved. Then, following O’Donnell and Coelli (Reference O’Donnell and Coelli2005), the following prior is used:

(30) \begin{equation} p\left ( \Sigma ^{-1}\right ) \propto \Sigma \end{equation}

which implies that $\Sigma ^{-1}$ is fully determined by the likelihood function—see the conditional posterior density for $\Sigma ^{-1}$ in equation (36).

As noted by Van den Broeck et al. (Reference Van den Broeck, Koop, Osiewalski and Steel1994), models based on exponential distribution are reasonably robust to changes in prior assumptions about the parameters. Therefore, an exponential distribution is used for the technical inefficiency $u_{it}$ with an unknown parameter $\lambda$ following Koop and Steel (Reference Koop, Steel, Koop and Steel2003); $u_{it}\sim i.i.d.\exp (\lambda ^{-1})$ . Since the exponential distribution is a gamma distribution when its first parameter equals one, the prior for $u_{it}$ can be written as:

(31) \begin{equation} p\left ( u_{it}\mid \lambda ^{-1}\right ) =f_{\textrm{Gamma}}\left ( u_{it}\mid 1,\lambda ^{-1}\right ). \end{equation}

According to Fernandez et al. (Reference Fernandez, Osiewalski and Steel1997), to obtain a proper posterior, a proper prior for the parameter $\lambda$ should be used. Therefore, the parameter $\lambda$ is assumed to have independent exponential prior with mean equals to $-1/\ln \tau ^{\ast }$ following Van den Broeck et al. (Reference Van den Broeck, Koop, Osiewalski and Steel1994). The prior independence of $\lambda$ leads to marginally prior independent of inefficiencies:

(32) \begin{equation} p(\lambda ^{-1})=f_{\textrm{Gamma}}\left ( \lambda ^{-1}\mid 1,-\ln \tau ^{\ast }\right ), \end{equation}

where $\tau ^{\ast }$ is the prior estimate of the mean of the technical efficiency distribution—see, for example, Koop et al. (Reference Koop, Osiewalski and Steel1997) and O’Donnell and Coelli (Reference O’Donnell and Coelli2005). Our best prior knowledge of the efficiency of US banks is the mean efficiency value of 0.4583 for DTDF with a prespecified directional vector, and 0.9431 for DTDF with a cost-optimal directional vector reported by Malikov et al. (Reference Malikov, Kumbhakar and Tsionas2016) who apply a Bayesian DTDF-cost system approach to large US commercial banks for 2001–2010 period. The mean output technical efficiency value of 0.9279 is reported by Feng and Serletis (Reference Feng and Serletis2010) who apply a Bayesian output distance function to US large banks for 2000–2005 period. To our knowledge, the input technical efficiency is not reported by any US banking study over a comparable period. However, the mean input technical efficiency value of 0.690 for all banks, 0.707 for small banks, and 0.735 for large banks are reported by Marsh et al. (Reference Marsh, Featherstone and Garrett2003) who apply a Bayesian input distance function to US commercial banks during 1990–2000. Since changing $\tau ^{\ast }$ changes the prior moments, various values of $\tau ^{\ast }$ within its possible range is experimented to assess the sensitivity of the results to changes to $\tau ^{\ast }$ . The results are the same up to the number of digits presented in Section 6, implying that the results are very robust to large changes in $\tau ^{\ast }$ .

To account for heterogeneity in the directional vectors for banks, prior distribution for $g_{i}$ is specified as a normal prior distribution with mean $G_{0}=1$ and a large variance:

(33) \begin{equation} p(g_{i})\sim N\!\left ( G_{0},\Omega _{G}\right ), \end{equation}

where $G_{0}$ is a vector of ones, and $\Omega _{G}$ is a diagonal matrix with $10^{4}$ in diagonal elements. The directional vector that projects any inefficient bank to the cost (revenue or profit) minimizing (maximizing) benchmark does not impose any sign restrictions on the adjustments of inputs and outputs. Therefore, these directional vectors may have negative components such that inputs are expanded, or outputs are contracted to reach the frontier at the cost (revenue or profit) minimizing (maximizing) benchmark—see, for example, Zofio et al. (Reference Zofio, Pastor and Aparicio2013) for a profit-optimal directional vector and Atkinson et al. (Reference Atkinson, Primont and Tsionas2018) for a cost and a profit-optimal directional vectors.

Using the priors in (29)–(33), and assuming that the prior distributions of the parameters are independent, the joint prior probability density function is therefore

(34) \begin{equation} f\left ( \beta,\Sigma ^{-1},u_{it},\lambda ^{-1},g_{i}\right ) =p(\beta )p\!\left ( \Sigma ^{-1}\right ) p\!\left ( u_{it}\mid \lambda ^{-1}\right ) p(\lambda ^{-1})p(g_{i}). \end{equation}

4.2. Full Conditional Posterior Distributions

Let $\Gamma =\left ( \beta,\Sigma ^{-1},u_{it},\lambda ^{-1},g_{i}\right )$ denotes all the parameters of the model, and $\Gamma _{-a}$ denotes all parameters other than $a$ . To derive the likelihood function, the Jacobian transformation matrix from the vector of random errors to the endogenous variables (inputs, outputs, or all the inputs and outputs) for the DIDF-cost, DODF-revenue, or DTDF-profit system is defined as $J_{it}\left ( g_{i},\beta \right ) =\frac{\partial \left ( v_{I,it}-u_{I,it},v_{C,it}\right ) }{\partial \left ( x_{1,it},x_{2,it}\right ) }$ , $J_{it}\!\left ( g_{i},\beta \right ) =\frac{\partial \left ( v_{O,it}-u_{O,it},v_{R,it}\right ) }{\partial \left ( y_{1,it},y_{2,it}\right ) }$ , or $J_{it}\!\left ( g_{i},\beta \right ) =\frac{\partial \left ( v_{T,it}-u_{T,it},v_{\pi w_{1},it},v_{\pi w_{2},it},v_{\pi p_{2},it}\right ) }{\partial \left ( y_{1,it},x_{1,it},x_{2,it},y_{2,it}\right ) }$ , respectively. Applying the Jacobian transformation, the likelihood function of $Y$ , given $\Gamma$ is

(35) \begin{equation} L\left ( Y\mid \Gamma \right ) =\left [ \underset{i=1}{\overset{K}{\Pi }}\underset{t=1}{\overset{T}{\Pi }}\,f_{\textrm{Normal}}\left ( Y_{it}\mid R_{it}\left ( g_{it}\right ) \beta -u_{it}\iota,I\otimes \Sigma \right ) \right ] \underset{i=1}{\overset{K}{\Pi }}\underset{t=1}{\overset{T}{\Pi }}\left \vert \det \left ( J_{it}\!\left ( g_{i},\beta \right ) \right ) \right \vert. \end{equation}

Using Bayes’ theorem and combining the likelihood function in (35) and the joint prior distributions in (34), the full conditional posterior distributions for all the parameters are found to be

(36) \begin{equation} p\left ( \Sigma ^{-1}\mid Y,\Gamma _{-\Sigma ^{-1}}\right ) \propto f_{\textrm{Gamma}}\left ( \Sigma ^{-1}\mid \frac{KT}{2},\frac{1}{2}\left ( Q_{it}+u_{it}\iota \right ) ^{\prime }\left ( Q_{it}+u_{it}\iota \right ) \right ) \text{,} \end{equation}

(37) \begin{equation} p\left ( \beta \mid Y,\Gamma _{-\beta }\right ) \propto f_{\textrm{Normal}}\left ( \beta \mid Dd,D\right ) \underset{i=1}{\overset{K}{\Pi }}\underset{t=1}{\overset{T}{\Pi }}\left \vert \det\!\left ( J_{it}\!\left ( g_{i},\beta \right ) \right ) \right \vert I\left ( \beta \in S_{j}\left ( g_{i}\right ) \right ) \text{,} \end{equation}

(38) \begin{equation} p\left ( \lambda ^{-1}\mid Y,\Gamma _{-\lambda ^{-1}}\right ) \propto f_{\textrm{Gamma}}\left ( \lambda ^{-1}\mid KT+1,u^{\prime }\iota _{KT}-\ln \tau ^{\ast }\right ) \text{,} \end{equation}

(39) \begin{equation} p\left ( u_{it}\mid Y,\Gamma _{-u_{it}}\right ) \propto f_{\textrm{Normal}}\left ( u_{it}\mid -\frac{\left ( Q_{it}^{\prime }\left ( g_{i}\right ) \Sigma ^{-1}\iota +\lambda ^{-1}\right ) }{\iota ^{\prime }\Sigma ^{-1}\iota },\frac{1}{\iota ^{\prime }\Sigma ^{-1}\iota }\right ) I\left ( u_{it}\geq 0\right ) \text{,} \end{equation}

(40) \begin{align} p\left ( g_{i}\mid Y,\Gamma _{-g_{i}}\right ) & \propto \left [ \underset{i=1}{\overset{K}{\Pi }}f_{\textrm{Normal}}\left ( Y_{i}\mid R_{i}\left ( g_{i}\right ) \beta -u_{i}\iota,I\otimes \Sigma \right ) \right ] \underset{i=1}{\overset{K}{\Pi }}\underset{t=1}{\overset{T}{\Pi }}\left \vert \det \left ( J_{it}\left ( g_{i},\beta \right ) \right ) \right \vert \nonumber \\[3pt] & \times \underset{i=1}{\overset{K}{\Pi }}f_{\textrm{Normal}}\left ( g_{i}\mid G_{0},\Omega _{G}\right ) \underset{i=1}{\overset{K}{\Pi }}I\left ( \beta \in S_{j}\left ( g_{i}\right ) \right ), \end{align}

where $D=\left ( R_{it}\left ( g_{i}\right ) ^{\prime }\left ( I\otimes \Sigma \right ) ^{-1}R_{it}\left ( g_{i}\right ) +\Omega _{\beta }^{-1}\right ) ^{-1}$ , $d=R_{it}\!\left ( g_{i}\right ) ^{\prime }\left ( I\otimes \Sigma \right ) ^{-1}\left ( Y_{it}+u_{it}\iota \right ) +\Omega _{\beta }^{-1}\beta _{0}$ , and $Q_{it}=Y_{it}-R_{it}\left ( g_{i}\right ) \beta$ . $I\left ( u_{it}\geq 0\right )$ is an indicator function that takes the value 1 if the constraint $u_{it}\geq 0$ is satisfied and 0 otherwise.

Bayesian estimation for a single-equation stochastic directional distance function without additional first-order condition equations and with a prespecified directional vector $g$ $=\left ( -1,1\right )$ or $g$ $=\left ( -x,y\right )$ can be implemented by setting $\underset{i=1}{\overset{K}{\Pi }}\underset{t=1}{\overset{T}{\Pi }}\left \vert \det \left ( J_{it}\!\left ( g_{i},\beta \right ) \right ) \right \vert =1$ and setting the relevant elements of $g$ equal to ones or $g_{x_{1i}}=x_{1i}$ , $g_{x_{2i}}=x_{2i}$ , $g_{y_{2i}}=y_{2i}$ , and normalizing the relevant directional vector. Bayesian estimation without theoretical regularity constraints can be implemented by setting $I\left ( \beta \in S_{j}\left ( g_{i}\right ) \right )$ in (37) and (40) equal to 1 and then drawing sequentially from the full conditional posteriors in (36)–(40).

4.3. Estimating the Interactive Effect

In the Bayesian framework, Koop et al. (Reference Koop, Osiewalski and Steel1997) propose a model where a time-invariant inefficiency is assumed to be exponentially distributed with producer-specific mean inefficiencies $\lambda _{i}$ and independent exponential priors; $u_{i}\sim i.i.d.\exp (\lambda _{i}^{-1})$ where $\lambda _{i}=\exp \left ( Z_{i}^{\prime }\delta \right )$ . Following Koop et al. (Reference Koop, Osiewalski and Steel1997), the inefficiency term $u_{T,it}$ , can be specified to be time-variant inefficiency by including bank-specific time-varying covariates in the parameter of an exponential distribution as $u_{T,it}\sim i.i.d.\exp (\lambda _{it}^{-1})$ , and $\lambda _{it}=\exp \left ( Z_{it}^{\prime }\delta \right )$ , where $\delta$ is an unknown vector of coefficients (including the intercept) to be estimated. The prior for $u_{T,it}$ can be written as:

(41) \begin{equation} p\left ( u_{T,it},\delta \mid Z\right ) =f_{\textrm{Gamma}}\left ( u_{T,it}\mid 1,\exp \left ( Z_{it}^{\prime }\delta \right ) \right ). \end{equation}

Following Koop et al. (Reference Koop, Osiewalski and Steel1997), the parameter vector $\delta$ is assumed to have a proper prior independent of the other parameters. A normal prior distribution with mean $\delta _{0}$ and variance $\Omega _{\delta }$ for $\delta$ is used:

(42) \begin{equation} p(\delta )\sim N(\delta _{0},\Omega _{\delta }), \end{equation}

where $\delta _{0}$ is a vector of zeros and $\Omega _{\delta }$ is a diagonal matrix with $10^{4}$ in diagonal elements. Note that by conditioning on $Y$ and $Z$ , bank-specific variables $Z$ are allowed to be correlated with the variables describing the frontier $Y$ . The full conditional posterior distributions for $\delta$ and $u_{T,it}$ are found to be

(43) \begin{equation} p\left ( \delta \mid Y,\Gamma _{-\delta }\right ) \propto f_{\textrm{Normal}}\left ( \delta \mid \frac{\delta _{0}\Omega _{\delta }^{-1}\iota -Z_{it}^{\prime }u_{T,it}}{\iota ^{\prime }\Omega _{\delta }^{-1}\iota },\frac{1}{\iota ^{\prime }\Omega _{\delta }^{-1}\iota }\right ) \text{,} \end{equation}

(44) \begin{equation} p\left ( u_{T,it}\mid Y,\Gamma _{-u_{T,it}}\right ) \propto f_{\textrm{Normal}}\left ( u_{T,it}\mid -\frac{\left ( Q_{it}^{\prime }\left ( g_{i}\right ) \Sigma ^{-1}\iota +\mu _{it}\right ) }{\iota ^{\prime }\Sigma ^{-1}\iota },\frac{1}{\iota ^{\prime }\Sigma ^{-1}\iota }\right ) I\left ( u_{T,it}\geq \mu _{it}\right ), \end{equation}

where $Q_{it}\!\left ( g_{i}\right ) =Y_{it}-R_{it}\!\left ( g_{i}\right ) \beta$ , and $\mu _{it}=Z_{it}^{\prime }\delta$ . Using MCMC methods, posteriors for input and output inefficiencies that derived separately are then used as proposal distributions in a computationally efficient second stage. See Lunn et al. (Reference Lunn, Barrett, Sweeting and Thompson2013).

6.1. Imposing the Theoretical Regularity Conditions

As required by neoclassical microeconomic theory, the production technology has to satisfy the theoretical regularity conditions of monotonicity and curvature. Monotonicity requires that the directional distance function be nondecreasing in inputs and nonincreasing in outputs. Therefore, monotonicity conditions of the DTDF imply the following restrictions:

(45) \begin{align} \frac {\partial \vec {D}_{T}(.)}{\partial x_{n}}&=\alpha _{n}+\sum _{n^{\prime }=1}^{2}\alpha _{nn^{\prime }}x_{n^{\prime }}+\left [ \alpha _{nn}g_{x_{n}}+\alpha _{nn^{\prime }}g_{x_{n^{\prime }}}-\gamma _{n2}g_{y_{2}}\right ] y_{1}+\gamma _{n2}y_{2}+\delta _{tx_{n}}t\geq 0\text { }\left ( n=1,2\right ) \text {;} \nonumber\\[3pt]\frac{\partial \vec{D}_{T}(.)}{\partial y_{1}} & =\left [ \alpha _{1}g_{x_{1}}+\alpha _{2}g_{x_{2}}-\beta _{2}g_{y_{2}}-1\right ] \nonumber\\ &\quad +\left [ \alpha _{11}g_{x_{1}}^{2}+\alpha _{22}g_{x_{2}}^{2}+\beta _{22}g_{y_{2}}^{2}+\alpha _{12}g_{x_{1}}g_{x_{2}}-\gamma _{12}g_{x_{1}}g_{y_{2}}-\gamma _{22}g_{x_{2}}g_{y_{2}}\right ] y_{1}\nonumber\\ &\quad +\left [ \gamma _{12}g_{x_{1}}+\gamma _{22}g_{x_{2}}-\beta _{22}g_{y_{2}}\right ] y_{2}\nonumber\\ &\quad +\left [ \alpha _{11}g_{x_{1}}+\alpha _{12}g_{x_{2}}-\gamma _{12}g_{y_{2}}\right ] x_{1}\nonumber\\ & \quad + \left [ \alpha _{21}g_{x_{1}}+\alpha _{22}g_{x_{2}}-\gamma _{22}g_{y_{2}}\right ] x_{2}\nonumber\\ &\quad +\left [ \delta _{tx_{1}}g_{x_{1}}+\delta _{tx_{2}}g_{x_{2}}-\delta _{ty_{2}}g_{y_{2}}\right ] t\leq 0\text{;} \nonumber\\[3pt]\frac{\partial \vec{D}_{T}(.)}{\partial y_{2}}&=\beta _{2}+\left [ \gamma _{12}g_{x_{1}}+\gamma _{22}g_{x_{2}}-\beta _{22}g_{y_{2}}\right ] y_{1}+\beta _{22}y_{2}+\sum _{n=1}^{2}\gamma _{n2}x_{n}+\delta _{ty_{2}}t\leq 0\text{.} \end{align}

When $g_{x_{1}}=g_{x_{2}}=0$ , monotonicity conditions of the DTDF in (45) reduces to the monotonicity conditions of the DODF. Monotonicity conditions of the DIDF imply the following restrictions:

\begin{align*} \frac {\partial \vec {D}_{I}(.)}{\partial x_{1}}&=\left [ 1-\alpha _{2}g_{x_{2}}\right ] +\alpha _{22}g_{x_{2}}^{2}x_{1}-\alpha _{22}g_{x_{2}}x_{2}-\sum _{m=1}^{2}\gamma _{2m}g_{x_{2}}y_{m}-\delta _{tx_{2}}g_{x_{2}}t\geq 0\text {;}\\[3pt]\frac {\partial \vec {D}_{I}(.)}{\partial x_{2}}&=\alpha _{2}-\alpha _{22}g_{x_{2}}x_{1}+\alpha _{22}x_{2}+\sum _{m=1}^{2}\gamma _{2m}y_{m}+\delta _{tx_{2}}t\geq 0\text {;}\\[3pt]\frac {\partial \vec {D}_{I}(.)}{\partial y_{m}}&=\beta _{m}+\sum _{m^{\prime }=1}^{2}\beta _{mm^{\prime }}y_{m^{\prime }}-\gamma _{2m}g_{x_{2}}x_{1}+\gamma _{2m}x_{2}+\delta _{tym}t\leq 0\text { }\left ( m=1,2\right ) \text {.}\end{align*}

Curvature restrictions can be imposed by ensuring that every principal minor of the Hessian matrix of odd order (even order) is nonpositive (nonnegative)—see, for example, Morey (Reference Morey1986). However, the US banking industry is highly regulated at both the federal and state levels, and different states change their regulatory restrictions at different times. This implies that the curvature condition would involve a bordered Hessian matrix that accounts for those regulatory restrictions. However, quantifying all those regulatory restrictions in the US banking industry is not an easy task. Furthermore, Barnett (Reference Barnett2002) notes that the imposition of global curvature on the quadratic functional form may induce spurious violations of monotonicity. Therefore, directional distance functions are estimated subject to theoretical monotonicity only following Färe et al. (Reference Färe, Grosskopf, Noh and Weber2005).

The unit value, the observed input−output, and the optimal direction models are first estimated without imposing the monotonicity conditions. However, the monotonicity conditions with respect to labor and all outputs are violated for all models at most observations. Thus, to produce inference that is consistent with neoclassical microeconomic theory, all models are re-estimated with the monotonicity conditions imposed at each observation, by following the Bayesian procedure discussed in O’Donnell and Coelli (Reference O’Donnell and Coelli2005).

6.2. Technical Inefficiency Measures

Observation-specific posterior estimates of technical inefficiency are obtained from the posterior conditional mean of $u$ . The average technical inefficiency measures for each sample year from the three directional vectors regularity constrained models are summarized in Table 3.

Comparing technical inefficiency measures across the unit value, the observed input−output, and the optimal direction models, the prespecified unit value and observed input−output direction models which leave the endogeneity of inputs and outputs unaddressed produce lower estimates of technical inefficiency. This implies that models that ignore the endogeneity problem tend to underestimate bank inefficiency measures. This suggests the importance of managing the endogeneity issue to obtain unbiased and consistent estimates of the parameters of the production technology and the associated measures of inefficiency—see, for example, Atkinson and Primont (Reference Atkinson and Primont2002).

As can be seen in Table 3, the total average of input and output technical inefficiency measures are larger than the average overall technical inefficiency measures in the case of the unit value and the optimal direction models and smaller than the average overall technical inefficiency measures in the case of the observed input−output direction modelFootnote ³ . These results are true for all years, except for years 2014 and 2015 in the case of the unit value direction model. A possible explanation of this is the problem of averaging inefficiency measures for all banks with different sizes for each sample year and recommends future research in comparing these inefficiencies for banks in small, medium, and large size classes.

Furthermore, output technical inefficiency is on average larger than input technical inefficiency in the three directional vectors models. This finding is consistent with Berger et al. (Reference Berger, Hancock and Humphrey1993) and English et al. (Reference English, Grosskopf, Hayes and Yaisawarng1993), who find that output inefficiency measures are as large or larger than input inefficiency measures. That is, most of the technical inefficiency in the US banking is in the form of loss of production, rather than overuse of inputs. These results indicate that banks may have more control over inputs than over outputs.

6.3. Results on the Interactive Effects

Now, we explore the effect of the variables included in the interactive effect equation, input and output technical inefficiencies, and the term capturing the interactions between them, on overall technical inefficiency. These technical inefficiency measures are of interest for several reasons. First, input and output inefficiencies measure different concepts and may affect future bank outcomes through different channels in the economy. Second, they include the effects of the overuse of inputs and the loss of production of outputs that occur when banks are inefficient. Third, they include the effects of the interactions between input and output inefficiencies on the efficiency of the banks. For instance, if a certain input (output) is viewed as being relatively overused (underproduced), it is likely that outputs (inputs) that are intensive in using (producing) that input (output) will be overproduced (underused) relative to other outputs (inputs). The interactive effect term captures output (input) implications of any errors in the input usage (output production). Thus, the interactive effect contains more information than could be captured in an input (output)-oriented efficiency study which excludes output (input) effects of input usage (output production) errors.

To focus on the relationships among input, output, and overall technical inefficiencies obtained from the systems of equations, consisting of DTDF and the interactive effect equations without and with the profit-maximizing first-order conditions, the estimated parameters of the DIDF and DODF for the three directional vectors models are not discussed. The estimated parameters of the DTDF, their associated 95% Bayes intervals, and their SIF values from the unit value, the observed input−output, and the optimal directional vector regularity constrained models are summarized in Table 4.

As can be seen in Table 4, UDTDF, VDTDF, and ODTDF models show that both input and output technical inefficiencies have significant positive effects on the overall technical inefficiency. However, the IEIO, $\delta _{IO}$ , has a significant negative effect on the overall technical inefficiency. This result is robust to alternative directional vectors and model specifications. Banks with larger values of the IEIO tend to have a lower level of overall technical inefficiency which indicates that they are more efficient. This suggests that the overuse of inputs creates input technical inefficiency and has an effect on reducing (improving) output technical inefficiency (efficiency) and therefore improving overall technical efficiency. Intuitively, the overuse of inputs whether physical inputs involving overuse of labor or overuse of financial inputs involving overpayment of interest may encourage banks to produce more loans to pay salaries for its employees and interest rates on deposits. Similarly, the loss of production of outputs creates output technical inefficiency and has an effect on reducing (improving) input technical inefficiency (efficiency) and therefore improving overall technical efficiency. Intuitively, the loss of revenue due to the loss of production of loans may encourage banks to reduce the number of labor used in the production process or lower the interest rates paid on deposits.

The most clarifying insights come from comparing bank-specific input, output, and overall technical inefficiency measures over the years 2001−2015. Figure 3 show an example of the interactions between input and output technical inefficiencies obtained based on the unit value, the observed input−output, and the optimal directional vector models. It is apparent that the increase in the output technical inefficiency reflects on a reduction on the input technical inefficiency and vice versa. Therefore, the IEIO directly results in a decrease in overall technical inefficiency. That is, a loss of output creates a loss of revenue and has an effect on the use of inputs, while an overuse of input creates additional costs and has an effect on the production of outputs. This helps make our argument that input and output approaches are not an appropriate metric for measuring inefficiency in the presence of the interactive effect. Instead, using models that incorporate both input and output inefficiencies is superior to the standard input or output approach for measuring inefficiency, since it allows to credit input and output while simultaneously crediting the interactive effect.

Figure 3. Technical inefficiency measures based on the unit value, the observed input−output, and the optimal directional vectors.

Differences in the magnitude of the IEIO are observed when using exogenous and endogenous directional vectors. In particular, the IEIO obtained based on the unit value and the observed input−output directional vector models appear to be of low magnitudes. On the other hand, considering banks heterogeneity in the directional vector, the IEIO obtained based on the optimal directional vector model appears to be of higher magnitude. This implies that models that ignore banks heterogeneity can lead to wrong conclusions concerning the estimates obtained from the models. This finding is consistent with Mester (Reference Mester1997) and Greene (Reference Greene2005) who find that heterogeneity causes biased estimates obtained from the SFA. This suggests the importance of managing banks heterogeneity to obtain unbiased estimates of the parameters of the production technology and the associated measures of inefficiency. Table 2 presents the minimum, maximum, and mean of the estimates of the optimal directional parameters. Letting the data select the directional vectors produces estimates of the directional parameters with a range of variation across banks.

Table 2. Estimates of optimal directional parameters

6.4. Technical Change

The parameter estimates of technical change $\delta _{t}$ , and $\delta _{tt}$ obtained based on the unit value, the observed input−output, and the optimal directional vector models appear to be of high magnitude—See Table 4. Specifically, it indicates significant technological advancement by the US banking industry over the period 2001−2015, which seems realistic given the recent advances in information technologies and its effects on the US banking industry.

Table 3. Average technical inefficiency over time based on the unit value, observed input−output, and optimal directional regularity-constrained models

Table 4. Parameter estimates for the regularity-constrained UDTDF, VDTDF, and ODTDF models