Imposing Regularity Conditions to Measure Banks’ Productivity Changes in Taiwan Using a Stochastic Approach

Abstract

This paper develops a stochastic approach to impose regularity properties on a directional output distance function (DODF) and an output distance function, which can be estimated by maximum likelihood. We use the resulting parameter estimates to evaluate efficiency and total factor productivity (TFP) growth for Taiwan’s commercial banks over the period 2002–2015 and claim that the failure of considering the regularity restrictions and the exclusion of undesirables lead to miscalculated efficiency measures and productivity gains. The outcomes from the regularity-constrained DODF reveal that almost all data-points satisfy the regularity properties, that the managerial abilities of the banks improve after the subprime crisis of 2007, and that the sample banks’ TFP grow at an average rate of 1.93% per annum, whereby technical change is the driving force. However, our estimates show downward trends in the growth rate of TFP and technical change.

Appendix

1.1 Imposition of Monotonicity and Curvature on an ODF

We re-define the vector of y as $y = \left( {y_{1} ,y_{2} } \right)^{\prime } \in R_{ + }^{2}$, because undesirables are excluded from ODF. The vector x is intact. We define the output set $S(x)$ as:

$$S(x) = \left\{ {y|y{\text{ can be produced by }}x} \right\}.$$

The output distance function $D_{o} \left( {y,x,t} \right)$ can be written as:

$$D_{o} \left( {y,x,t} \right) = min\left\{ {\delta |0 < \delta \le 1, y/\delta \in S\left( x \right)} \right\}.$$

(24)

A value of $D_{o} \left( {y,x,t} \right)$ equaling unity means that the bank is already producing on the efficient frontier, while a value of $D_{o} \left( {y,x,t} \right)$ less than unity reveals that the bank is technically inefficient due to managerial inability. Following Orea (2002), O’Donnell and Coelli (2005), and Feng and Serletis (2010, 2014), we specify the translog functional form for ODF:

$$\begin{aligned} \ln D_{o} (y,x,t) & = a_{0} + \sum\limits_{i = 1}^{2} {a_{i} \ln y_{i} } + \frac{1}{2}\sum\limits_{i = 1}^{2} {\sum\limits_{k = 1}^{2} {a_{ik} \ln y_{i} \ln y_{k} + } } \sum\limits_{n = 1}^{3} {b_{n} } \ln x_{n} + \frac{1}{2}\sum\limits_{n = 1}^{3} {\sum\limits_{j = 1}^{3} {b_{nj} \ln x_{n} \ln x_{j} } } \\ & \quad + \delta_{\tau } t + \frac{1}{2}\delta_{\tau \tau } t^{2} + \sum\limits_{n = 1}^{3} {\sum\limits_{i = 1}^{2} {g_{ni} \ln x_{n} \ln y_{i} } } + \sum\limits_{i = 1}^{2} {\delta_{\tau i} t\ln y_{i} } + \sum\limits_{n = 1}^{3} {\delta_{\tau n} t\ln x_{n} } \, \\ \end{aligned}$$

(25)

After imposing the homogeneity and symmetrical properties and appending a statistical noise term, we transform ODF into a regression equation:

$$\begin{aligned} - \ln y_{1} & = a_{0} + a_{2} \ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right) + \frac{1}{2}a_{22} \left[ {\ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right)} \right]^{2} + b_{1} \ln x_{1} + b_{2} \ln x_{2} + b_{3} \ln x_{3} + \frac{1}{2}b_{11} \left( {\ln x_{1} } \right)^{2} \\ & \quad + \frac{1}{2}b_{22} \left( {\ln x_{2} } \right)^{2} + \frac{1}{2}b_{33} \left( {\ln x_{3} } \right)^{2} + b_{12} \ln x_{1} \ln x_{2} + b_{13} \ln x_{1} \ln x_{3} + b_{23} \ln x_{2} \ln x_{3} \\ & \quad + \delta_{\tau } t + \frac{1}{2}\delta_{\tau \tau } t^{2} + g_{12} \ln x_{1} \ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right) + g_{22} \ln x_{2} \ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right) + g_{32} \ln x_{3} \ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right) \\ & \quad + \delta_{\tau 2} t\ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right) + \delta_{\tau 1} t\ln x_{1} + \delta_{\tau 2} t\ln x_{2} + \delta_{\tau 3} t\ln x_{3} { + }V_{1} { + }U_{1} \\ \end{aligned}$$

(26)

where $U_{1} = - \ln D_{o} (y,x,t)\sim\left| {N\left( {0,\sigma_{U1}^{2} } \right)} \right|$ is a one-sided error signifying technical inefficiency, $V_{1} \sim N\left( {0,\sigma_{V1}^{2} } \right)$ is the random disturbance, and $U_{1}$ and $V_{1}$ are assumed to be statistically independent.

1.2 Monotonicity and Curvature Constraints

Monotonicity requires that $D_{o} \left( {y,x,t} \right)$ be non-decreasing in outputs and non-increasing in inputs—that is:

$$\frac{{\partial D_{o} }}{{\partial y_{2} }} = \frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }}\frac{{D_{o} }}{{y_{2} }} \, \ge \, 0\;{\text{and}}\;\frac{{\partial D_{o} }}{{\partial x_{n} }} = \frac{{\partial \ln D_{o} }}{{\partial \ln x_{n} }}\frac{{D_{o} }}{{x_{n} }} \, \le \, 0,\quad n = 1,2,3,$$

(27)

or, equivalently:

$$\frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }} \ge 0\;{\text{and}}\;\frac{{\partial \ln D_{o} }}{{\partial \ln x_{n} }} \le 0,\quad n = 1,2,3,$$

(28)

since $D_{o} /y_{2}$ and $D_{o} /x_{n}$ are positive. The property $\partial D_{o} /\partial y_{2} \ge 0$ implies that a bank’s technical efficiency does not diminish when it can produce more of an output, say $y_{2}$, after employing a given input mix. The property $\partial D_{o} /\partial x_{n} \le 0$ implies that a bank’s technical efficiency does not rise when it hires more of an input, say $x_{n}$, to produce the same output levels.

We re-write (28) for output $y_{2}$ as:

$$\frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }} = a_{2} + a_{22} \ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right) + g_{12} \ln x_{1} + g_{22} \ln x_{2} + g_{32} \ln x_{3} + \delta_{\tau 2} t \, \ge 0.$$

Since the ODF must be homogeneous in $y_{1}$ and $y_{2}$, we have:

$$\frac{{\partial \ln D_{o} }}{{\partial \ln y_{1} }} = 1 - \frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }} \ge 0\;{\text{and}}\;\frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }} \le 1.$$

Translating the above inequalities into equalities, we obtain:

$$a_{2} + a_{22} \ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right) + g_{12} \ln x_{1} + g_{22} \ln x_{2} + g_{32} \ln x_{3} + \delta_{\tau 2} t \, = V_{2} + U_{2}$$

(29)

$$a_{2} - 1 + a_{22} \ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right) + g_{12} \ln x_{1} + g_{22} \ln x_{2} + g_{32} \ln x_{3} + \delta_{\tau 2} t \, = V_{3} - U_{3} .$$

(30)

Similarly, the monotonicity conditions for the three inputs can be expressed as:

$$b_{1} + b_{11} \ln x_{1} + b_{12} \ln x_{2} + b_{13} \ln x_{3} + g_{12} \ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right) + \delta_{\tau 1} t = V_{4} - U_{4} ,$$

(31)

$$b_{2} + b_{22} \ln x_{2} + b_{12} \ln x_{1} + b_{23} \ln x_{3} + g_{22} \ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right) + \delta_{\tau 2} t = V_{5} - U_{5} ,$$

(32)

$$b_{3} + b_{33} \ln x_{3} + b_{13} \ln x_{1} + b_{23} \ln x_{2} + g_{32} \ln \left( {\frac{{y_{2} }}{{y_{1} }}} \right) + \delta_{\tau 3} t = V_{6} - U_{6} .$$

(33)

Here, $U_{n} \sim \left| {N\left( {0,\sigma_{Un}^{2} } \right)} \right|$ and $V_{n} \sim N\left( {0,\sigma_{Vn}^{2} } \right)$, n = 2,…, 6, are respectively one-sided and two-sided errors and are statistically independent. The presence of $V_{n}$ in (29)–(33) makes the individual equations be regression equations and captures random shocks.

Curvature requires that ODF be convex in outputs and quasi-convex in inputs. See, for example, Färe and Grosskopf (1994) and O’Donnell and Coelli (2005). For ODF to be convex in outputs it is sufficient that all the principal minors of the Hessian matrix are non-negative. In our two-output case this requires that:

$$\frac{{\partial^{2} D_{o} }}{{\partial y_{2}^{2} }} = \frac{{\partial^{2} \ln D_{o} }}{{\partial \left( {\ln y_{2} } \right)^{2} }}\frac{1}{{y_{2} }}\frac{{D_{o} }}{{y_{2} }} + \left( {\frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }}} \right)^{2} \frac{{D_{o} }}{{y_{2}^{2} }} - \frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }}\frac{{D_{o} }}{{y_{2}^{2} }} \ge 0,$$

or, equivalently:

$$a_{22} + \frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }}\left( {\frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }} - 1} \right) \ge \, 0,$$

(34)

since $D_{o} /y_{2}^{2} \ge 0$. The necessary condition of (34) is $a_{22} \ge 0$, because:

$$\frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }}\left( {\frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }} - 1} \right) \le 0.$$

Equation (28) can be transformed into an equality:

$$a_{22} + \frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }}\left( {\frac{{\partial \ln D_{o} }}{{\partial \ln y_{2} }} - 1} \right) = \, U_{7} + V_{7} .$$

(35)

Quasi-convexity in inputs will be achieved if and only if all the principal minors of the following bordered Hessian matrix are non-positive:

$$F = \left( {\begin{array}{*{20}l} 0 \hfill & \quad {f_{1} } \hfill & \quad {f_{2} } \hfill & \quad {f_{3} } \hfill \\ {f_{1} } \hfill & \quad {f_{11} } \hfill & \quad {f_{12} } \hfill & \quad {f_{13} } \hfill \\ {f_{2} } \hfill & \quad {f_{12} } \hfill & \quad {f_{22} } \hfill & \quad {f_{23} } \hfill \\ {f_{3} } \hfill & \quad {f_{13} } \hfill & \quad {f_{23} } \hfill & \quad {f_{33} } \hfill \\ \end{array} } \right) ,$$

where $f_{n} = \partial D_{o} /\partial x_{n} = \left( {D_{o} /x_{n} } \right)\partial \ln D_{o} /\partial \ln x_{n}$, n = 1, 2, 3; $f_{mn} = \partial^{2} D_{o} /\partial x_{m} \partial x_{n}$, m, n = 1, 2, 3. There are seven principal minors in total to be considered. Among them, the following three conditions obviously hold for certainty and hence can be ignored, i.e.:

$$F_{11} = \left| {\left( {\begin{array}{*{20}c} 0 & {f_{1} } \\ {f_{1} } & {f_{11} } \\ \end{array} } \right)} \right| = - \left( {f_{1} } \right)^{2} \le 0,$$

$$F_{22} = \left| {\left( {\begin{array}{*{20}c} 0 & {f_{2} } \\ {f_{2} } & {f_{22} } \\ \end{array} } \right)} \right| = - \left( {f_{2} } \right)^{2} \le 0,$$

$$F_{33} = \left| {\left( {\begin{array}{*{20}c} 0 & {f_{3} } \\ {f_{3} } & {f_{33} } \\ \end{array} } \right)} \right| = - \left( {f_{3} } \right)^{2} \le 0.$$

However, the remaining four conditions must be considered:

$$F_{312} = \left| {\left( {\begin{array}{*{20}l} 0 \hfill & {f_{1} } \hfill & {f_{2} } \hfill \\ {f_{1} } \hfill & {f_{11} } \hfill & {f_{12} } \hfill \\ {f_{2} } \hfill & {f_{12} } \hfill & {f_{22} } \hfill \\ \end{array} } \right)} \right| = 2f_{1} f_{12} f_{2} - \left( {f_{2} } \right)^{2} f_{11} - \left( {f_{1} } \right)^{2} f_{22} \le 0,$$

$$F_{313} = \left| {\left( {\begin{array}{*{20}l} 0 \hfill & {f_{1} } \hfill & {f_{3} } \hfill \\ {f_{1} } \hfill & {f_{11} } \hfill & {f_{13} } \hfill \\ {f_{3} } \hfill & {f_{13} } \hfill & {f_{33} } \hfill \\ \end{array} } \right)} \right| = 2f_{1} f_{13} f_{3} - \left( {f_{3} } \right)^{2} f_{11} - \left( {f_{1} } \right)^{2} f_{33} \le 0,$$

$$F_{323} = \left| {\left( {\begin{array}{*{20}l} 0 \hfill & {f_{2} } \hfill & {f_{3} } \hfill \\ {f_{2} } \hfill & {f_{22} } \hfill & {f_{23} } \hfill \\ {f_{3} } \hfill & {f_{23} } \hfill & {f_{33} } \hfill \\ \end{array} } \right)} \right| = 2f_{2} f_{23} f_{3} - \left( {f_{3} } \right)^{2} f_{22} - \left( {f_{2} } \right)^{2} f_{33} \le 0,$$

$$F = \left| {\left( {\begin{array}{*{20}l} 0 \hfill & {f_{1} } \hfill & {f_{2} } \hfill & {f_{3} } \hfill \\ {f_{1} } \hfill & {f_{11} } \hfill & {f_{12} } \hfill & {f_{13} } \hfill \\ {f_{2} } \hfill & {f_{12} } \hfill & {f_{22} } \hfill & {f_{23} } \hfill \\ {f_{3} } \hfill & {f_{13} } \hfill & {f_{23} } \hfill & {f_{33} } \hfill \\ \end{array} } \right)} \right| = - f_{1} \left| {\begin{array}{*{20}c} {f_{1} } & {f_{12} } & {f_{13} } \\ {f_{2} } & {f_{22} } & {f_{23} } \\ {f_{3} } & {f_{23} } & {f_{33} } \\ \end{array} } \right| + f_{2} \left| {\begin{array}{*{20}c} {f_{1} } & {f_{11} } & {f_{13} } \\ {f_{2} } & {f_{12} } & {f_{23} } \\ {f_{3} } & {f_{13} } & {f_{33} } \\ \end{array} } \right| - f_{3} \left| {\begin{array}{*{20}c} {f_{1} } & {f_{11} } & {f_{12} } \\ {f_{2} } & {f_{12} } & {f_{22} } \\ {f_{3} } & {f_{13} } & {f_{23} } \\ \end{array} } \right| \le 0.$$

Let $S_{n} = \partial \ln D_{o} /\partial \ln x_{n}$, n = 1, 2, 3. We simplify and re-formulate the foregoing four inequalities into equalities:

$$2S_{1} S_{2} h_{12} - S_{2}^{2} h_{11} - S_{1}^{2} h_{22} = V_{8} - U_{8} ,$$

(36)

$$2S_{1} S_{3} h_{13} - S_{3}^{2} h_{11} - S_{1}^{2} h_{33} = V_{9} - U_{9} ,$$

(37)

$$2S_{2} S_{3} h_{23} - S_{3}^{2} h_{22} - S_{2}^{2} h_{33} = V_{10} - U_{10} ,$$

(38)

$$\begin{aligned} & - S_{1}^{2} h_{22} h_{33} - 2h_{12} h_{23} S_{1} S_{3} - 2S_{1} S_{2} h_{13} h_{23} + 2h_{13} h_{22} S_{1} S_{3} + 2h_{12} h_{33} S_{1} S_{2} + h_{23}^{2} S_{1}^{2} \hfill \\ & \quad + 2h_{11} h_{23} S_{2} S_{3} + h_{13}^{2} S_{2}^{2} - 2h_{12} h_{13} S_{2} S_{3} - h_{11} h_{33} S_{2}^{2} - h_{11} h_{22} S_{3}^{2} + h_{12}^{2} S_{3}^{2} = V_{11} - U_{11} . \hfill \\ \end{aligned}$$

(39)

Here, $h_{11} = S_{1}^{2} + b_{11} - S_{1}$, $h_{22} = S_{2}^{2} + b_{22} - S_{2}$, $h_{33} = S_{3}^{2} + b_{33} - S_{3}$, $h_{12} = S_{1} S_{2} + b_{12}$,$h_{13} = S_{1} S_{3} + b_{13}$, and $h_{23} = S_{2} S_{3} + b_{23}$.

As for the error components in (35)–(39) we assume that $U_{n} \sim \left| {N\left( {0,\sigma_{Un}^{2} } \right)} \right|$ and $V_{n} \sim N\left( {0,\sigma_{Vn}^{2} } \right)$, n = 7,…, 11, are respectively one-sided and two-sided errors and are statistically independent. The presence of $V_{n}$ in the five equations makes the individual equations be regression equations and captures statistical noise.

1.3 Estimation Procedure

Equations (26), (29)–(33), and (35)–(39) form a system of regression equations with composed errors. Since (29) and (30) are linearly dependent we arbitrarily delete (30) from the system of equations and suggest estimating the remaining ten equations simultaneously by ML. In this manner, we can similarly impose monotonicity and curvature on the ODF of (26) to the usage of a Bayesian approach. It is noteworthy that here we impose five more curvature restrictions on ODF compared to DODF, whose curvature conditions involve merely unknown parameters.

Following the case of DODF, we assume that all of the eleven composed errors, either v + u or v − u, in those equations are statistically independent. Their individual pdf’s are similar to (14) in the text and the corresponding likelihood function can be readily derived. We therefore omit their derivation. After getting the coefficient estimates, we apply the following formula, proposed by Battese and Coelli (1988), to directly calculate the measure of technical efficiency:

$$TE_{o} = E\left( {e^{{ - U_{1} }} |\omega_{1} } \right) = \frac{{\varPhi \left( {\frac{{\mu_{*1} - \sigma_{*1}^{2} }}{{\sigma_{*} }}} \right)}}{{\varPhi \left( {\frac{{\mu_{*1} }}{{\sigma_{*} }}} \right)}}\exp \left( {0.5\sigma_{*1}^{2} - \mu_{*1} } \right),$$

(40)

where $\omega_{1} = V_{1} + U_{1}$, $\sigma_{1}^{2} = \sigma_{U1}^{2} + \sigma_{V1}^{2}$, $\mu_{*1} = - \omega_{1} \sigma_{U1}^{2} /\sigma_{1}^{2}$, and $\sigma_{*1}^{2} = \sigma_{U1}^{2} \sigma_{V1}^{2} /\sigma_{1}^{2}$. Term $TE_{o}$ must lie between 0 and 1.

An analogous caveat to the case of DODF is worth citing, i.e., the resulting set of slope coefficient estimates may not be able to ensure that all monotonicity and curvature conditions are satisfied for all data points. This is attributed to the addition of the disturbances V_j’s, j = 2,…, 11, to Eqs. (29)–(33) and (35)–(39) and appears to be an advantage over the Bayesian approach. The number of violating observations is expected to be relatively small. We shall discuss this in the empirical study section below.