Why fully efficient banks matter? A nonparametric stochastic frontier approach in the presence of fully efficient banks

Abstract

A common assumption in the banking stochastic performance literature refers to the non-existence of fully efficient banks. This paper relaxes this strong assumption and proposes an alternative semiparametric zero-inefficiency stochastic frontier model. Specifically, we consider a nonparametric specification of the frontier whilst maintaining the parametric specification of the probability of fully efficient bank. We propose an iterative local maximum likelihood procedure that achieves the optimal convergence rates of both nonparametric frontier and the parameters contained in the probability of fully efficient bank. In an empirical application, we apply the proposed model and the estimation procedure to a global banking data set to derive new corrected measures of bank performance and productivity growth across the world. The results show that there is variability across regions, and the probability of fully efficient bank is mostly affected by bank-specific variables that are related to bank’s risk-taking attitude, whereas country-specific variables, such as inflation, also have an effect.

Appendices

Appendix A: proofs of the theorems

Let $\tilde{\gamma }(.) = \{ \tilde{m}(.),\tilde{\sigma }^{2} (.),\tilde{\lambda }(.)\}^{'}$. Also let $\gamma (.) = \{ m(.),\sigma^{2} (.),\lambda (.)\}^{'}$ and $\alpha$ denote the true values.

Proof of Theorem 1

The proof of this theorem follows similarly to the proof of Theorem 1 of Tran and Tsionas (2016a) and Huang and Yao (2012). Thus, we only outline the key steps of the proof.

To derive the asymptotic properties of $\hat{\alpha }$, we first let

$$ \begin{aligned} \hat{\alpha }^{*} & = \sqrt n (\hat{\alpha } - \alpha ), \\ \ell \left( {\tilde{\gamma }(X_{i} ),\alpha ,Z_{i} ,Y_{i} } \right) & = \log f\left( {Y_{i} |\tilde{\gamma }(X_{i} ),\alpha ,Z_{i} } \right) \\ \ell \left( {\tilde{\gamma }(X_{i} ),\hat{\alpha } + n^{ - 1/2} \alpha^{*} ,Z_{i} ,Y_{i} } \right) & = \log f\left( {Y_{i} |\tilde{\gamma }(x_{i} ),\hat{\alpha } + n^{ - 1/2} \alpha^{*} ,Z_{i} } \right) \\ \end{aligned} $$

Then $\hat{\alpha }^{*}$ is the maximization of

$$ L_{n} \left( {\alpha^{*} } \right) = \sum\limits_{i = 1}^{n} {\left\{ {\ell \left( {\tilde{\gamma }(X_{i} ),\alpha + n^{ - 1/2} \alpha^{*} ,Z_{i} ,Y_{i} } \right) - \ell \left( {\tilde{\gamma }(X_{i} ),\alpha ,Z_{i} ,Y_{i} } \right)} \right\}} . $$

(A.1)

By using a Taylor series expansion and after some calculation, it yields

$$ L_{n} (\alpha^{*} ) = A_{n} \alpha^{*} + \frac{1}{2}\alpha^{*'} B_{n} \alpha^{*} + o_{p} (1), $$

(A.2)

where

$$ A_{n} = n^{ - 1/2} \sum\limits_{i = 1}^{n} {\frac{{\partial \ell (\tilde{\gamma }(X_{i} ),\alpha ,Z_{i} ,Y_{i} )}}{\partial \alpha }} $$

$$ B_{n} = n^{ - 1} \sum\limits_{i = 1}^{n} {\frac{{\partial^{2} \ell (\tilde{\gamma }(X_{i} ),\alpha ,Z_{i} ,Y_{i} )}}{{\partial \alpha \partial \alpha^{'} }}} . $$

Next we evaluate the terms $A_{n}$ and $B_{n}$. First, expanding $A_{n}$ around $\gamma (X_{i} )$, we obtain

$$ \begin{aligned} A_{n} & = n^{ - 1/2} \sum\limits_{i = 1}^{n} {\frac{{\partial \ell \left( {\gamma (X_{i} ),\alpha ,Z_{i} ,Y_{i} } \right)}}{\partial \alpha }} + n^{ - 1/2} \sum\limits_{i = 1}^{n} {\frac{{\partial^{2} \ell \left( {\gamma (X_{i} ),\alpha ,Z_{i} ,Y_{i} } \right)}}{{\partial \alpha \partial \gamma^{'} }}\left[ {\tilde{\gamma }(X_{i} ) - \gamma (X_{i} )} \right]} \\ & \,\,\,\,\, + O_{p} \left( {n^{ - 1/2} ||\tilde{\gamma }(.) - \gamma (.)||_{\infty }^{2} } \right) \\ & = n^{ - 1/2} \sum\limits_{i = 1}^{n} {\frac{{\partial \ell \left( {\gamma (X_{i} ),\alpha ,Z_{i} ,Y_{i} } \right)}}{\partial \alpha }} + D_{1n} + O_{p} \left( {n^{ - 1/2} ||\tilde{\gamma }(.) - \gamma (.)||_{\infty }^{2} } \right), \\ \end{aligned} $$

where the definition of $D_{1n}$ should be apparent. Following Tran and Tsionas (2016a, b), it can be shown that

$$ A_{n} = n^{ - 1/2} \sum\limits_{i = 1}^{n} {\left\{ {\frac{{\partial \ell (\gamma (X_{i} ),\alpha ,Z_{i} ,Y_{i} )}}{\partial \alpha } - I_{\alpha \gamma } (X_{i} )d(X_{i} ,Y_{i} ,Z_{i} )} \right\}} + o_{p} (1), $$

(A.3)

where $d(X,Y,Z)$ is the first $r \times r$ submatrix of $I_{\theta \theta }^{ - 1} (X)q_{\theta } (\theta (X),Z,Y)$. Similarly, for $B_{n}$, it can be shown that

$$ B_{n} = - E\left[ {I_{\alpha \alpha } (X)} \right] + o_{p} (1) = B + o_{p} (1). $$

(A.4)

Thus, from (A.2) in conjunction with (A.4), an application of quadratic approximation lemma [see, for example, Fan and Gijbels (1996, p. 210)] leads to

$$ \hat{\alpha }^{*} = B^{ - 1} A_{n} + o_{p} (1) $$

(A.5)

if $A_{n}$ is a sequence of stochastically bounded vectors. Consequently, the asymptotic normality of $\hat{\alpha }^{*}$ follows from that of $A_{n}$. Note that since $A_{n}$ is the sum of i.i.d. random vectors, it suffices to compute the mean and covariance matrix of $A_{n}$ and evoke the central limit theorem. To this end, from (A.3), we have

$$ E(A_{n} ) = n^{1/2} E\left\{ {\frac{\partial \ell (\gamma (X),\alpha ,Z,Y)}{\partial \alpha } - I_{\alpha \gamma } (X)d(X,Y,Z)} \right\}. $$

(A.6)

The expectation of each element of the first term on the right-hand side can be shown to be equal to 0, and further calculation shows that $E\left\{ {I_{\alpha \gamma } (X)d(X,Y,Z)} \right\} = 0$. Thus, $E(A_{n} ) = 0$. The variance of $A_{n}$ is ${\text{Var}}(A_{n} ) = {\text{Var}}\left\{ {\frac{{\partial \ell \left( {\gamma (x),\alpha ,Z,y} \right)}}{\partial \alpha } - I_{\alpha \gamma } (X)d(X,Y,Z)} \right\} =\Sigma$. By the central limit theorem, we obtain the desired result.□

Proof of Theorem 2

Recall that, given the estimate of $\hat{\alpha }$, $\hat{\gamma }(x)$ maximizes (7). By redefining appropriate notations:

$$ \begin{aligned} \eta (x_{0} ,X) &= \gamma_{0} (x_{0} ) +\Gamma _{1} (x_{0} )(X - x_{0} ), \\ \gamma^{*} &= \left( {n\left| H \right|} \right)^{1/2} \left\{ {\gamma - \gamma_{0} (x_{0} ),\left| H \right|(\gamma^{'} -\Gamma _{1} (x_{0} ))} \right\}^{'} , \\ \end{aligned} $$

then the proof follows directly from the proof of Theorem 2 of Tran and Tsionas (2016a, b). Thus, we omit it here for brevity.□

Appendix B: fully localized model

The discussion in Sect. 2 has been limited to the case where the probability of fully efficient firm $\pi (z)$ is assumed to have a logistic function. In this appendix, we extend the model to allow for a nonparametric function $\pi (z)$. We will consider two cases. In the first case, we assume that $Z = X$ and show how to estimate this model as well as discuss the asymptotic properties of the local MLE. In the second case where in general $Z \ne X$, we will briefly discuss only the estimation procedure but not the asymptotic properties since they are more complicated and beyond the scope of this paper.

Case 1: When $Z = X$

In this case, we first redefine the vector function $\theta (x) = (\pi (x),\gamma (x)^{'} )^{'}$ and for a given set point $x_{0}$ and $x$ in the neighbourhood of $x$, we approximate the function $\theta (x)$ by a linear function similar to (5),

$$ \theta (x_{0} ) \approx \theta_{0} (x_{0} ) +\Theta _{1} (x_{0} )(x - x_{0} ), $$

where $\theta_{0} (x_{0} )$ is a $(4 \times 1)$ vector and $\Phi_{1} (x_{0} )$ is a $(4 \times d)$ matrix of the first-order derivatives. Then the conditional local log-likelihood function is:

$$ L_{5n} \left( {\theta_{0} (x_{0} ),\Theta _{1} (x_{0} )} \right) = \sum\limits_{i = 1}^{n} {\left\{ {\log f\left( {Y_{i} ;\theta_{0} (x_{0} ) + \Theta_{1} (x_{0} )(X_{i} - x_{0} )} \right)} \right\}} K_{H} \left( {X_{i} - x_{0} } \right), $$

(B.1)

where the kernel function $K_{H} (X_{i} - x_{0} )$ is defined as before. Let $\hat{\theta }_{0} (x_{0} )$ denote the local maximizer of (B.1). Then the local MLE of $\theta (x)$ is given by $\hat{\theta }(x) = \hat{\theta }_{0} (x_{0} )$. To obtain the asymptotic property of $\hat{\theta }(.)$, we modify the following notations:

$$ \begin{aligned} q_{1} (\theta (x),Y) & = \partial L_{5} (\theta (x),Y)/\partial \theta ,\,\,\,\,\,q_{2} (\theta (x),Y) = \partial^{2} L_{5} (\theta (x),Y)/\partial \theta \partial \theta^{'} , \\ \\ I(x) & = - E\left\{ {q_{2} (\theta (X),Y)|X = x} \right\},\,\,{\text{and}}\,\,\,\Psi (u|x) = \int\limits_{Y} {q_{1} (\theta (x),Y)f(Y|\theta (u)){\text{d}}Y} . \\ \end{aligned} $$

Assumptions

B1: The support for $X$, denoted by ${\mathcal{X}}$, is compact subset of ${\mathbb{R}}^{d}$. Furthermore, the marginal density $f(x)$ of $X$ is twice continuously differentiable and positive for $x \in {\mathcal{X}}$.

B2: The unknown function $\theta (x)$ has continuous second derivatives, and in addition, $\sigma^{2} (x) > 0$ and $0 < \pi (x) < 1$ hold for all $x \in {\mathcal{X}}$.

B3: There exists a function ${\mathcal{M}}(y),$ with $E[{\mathcal{M}}(y)] < \infty$ such that for all $Y$, and all $\theta \in nbhd\,{\text{ of }}\theta (x)$, $|\partial L_{5} (\theta ,Y)/\partial \theta_{j} \partial \theta_{k} \partial \theta_{l} |\,\, < {\mathcal{M}}(y).$

B4: The following conditions hold for all $i$ and $j$:$E\{ |\partial L_{5} (\theta (x),Y)/\partial \theta_{j} |^{3} \} < \infty ,\,\,\,\,\,E\{ (\partial^{2} L_{5} (\theta (x),Y)/\partial \theta_{i} \partial \theta_{j} )^{2} \} < \infty$.

B5: The kernel function $K(.)$ has bounded support and satisfies:

$$ \begin{aligned} \left( {\int {K(u){\text{d}}u} } \right)I_{d} & = 1,\,\,\,\,\,\,\left( {\int {uK(u){\text{d}}u} } \right)I_{d} = 0,\,\,\,\,\,\left( {\int {u^{2} K(u){\text{d}}u} } \right)I_{d} < \infty , \\ \\ \left( {\int {K^{2} (u){\text{d}}u} } \right)I_{d} & < \infty ,\,\,\,\,\left( {\int {|K(u)|^{3} {\text{d}}u} } \right)I_{d} < \infty . \\ \end{aligned} $$

B6: $|H|\,\, \to 0,\,\,\,n|H|\,\, \to \infty ,{\text{ and }}n|H|^{5} = O(1){\text{ as }}n \to \infty .$

Proposition 1

Suppose that conditions (B1)–(B6) hold. Then it follows that

$$ (n|H|)^{1/2} \left\{ {\hat{\theta }(x) - \theta (x) - {\mathcal{B}}(x) + o(|H|)^{2} } \right\}{\mathop{\longrightarrow}\limits^{D}}N\left( {0,\kappa_{0} f^{ - 1} (x)I_{\theta \theta }^{ - 1} } \right), $$

where${\mathcal{B}}(x) = \frac{1}{2}\mu_{2} |H|^{2} I_{\theta \theta }^{ - 1} (z)\Psi ^{''} (x|x)$with$\kappa_{0}$and$\mu_{2}$being defined as in Sect. 2.

The proof of Proposition 1 is a straightforward extension of the proof of Theorem 2 in Huang et al. (2013) to the multivariate case, and hence, it is omitted.

Case 2: When $Z \ne X$

In this case, the local MLE is similar to case 1, albeit it is more complicated. To see this, let us once again redefine the vector function $\theta (z,x) = (\pi (z),\gamma (x)^{'} )^{'}$; then, for a given set points $z_{0}$ and $x_{0}$, approximate $\theta (z,x)$ linearly as before. Also, define the kernel function for $z$ as $W_{A} (Z_{i} ,z_{0} ) = |A|^{\, - 1} W(A^{ - 1} (Z_{i} - z_{0} ))$ where $W(v) = \prod\nolimits_{j = 1}^{r} {w(v_{j} )}$ with $w(.)$ being a univariate probability function, $A$ being a bandwidth matrix and $\left| A \right| = a_{1} a_{2} \ldots a_{r}$. Then the modified conditional local log-likelihood function can be written as:

$$\begin{aligned} &L_{6n} (\theta_{0} (z_{0} ,x_{0} ),\varTheta_{1} (z_{0} ,x_{0} ))\\ &\quad= \sum\limits_{i = 1}^{n} {\left\{ {\log f(Y_{i} ;\theta_{0} (z_{0} ,x_{0} ) + \varTheta_{1} (z_{0} ,x_{0} )(Z_{i} - z_{0} )(X_{i} - x_{0} ))} \right\}}\\ &\qquad\times W_{{A_{1} }} (Z_{i} - z_{0} )K_{{H_{1} }} (X_{i} - x_{0} ).\end{aligned} $$

(B.2)

Let $\theta^{*} (z_{0} ,x_{0} )$ be the maximizer of (14) where $\theta^{*} (z_{0} ,x_{0} ) = (\pi^{*} (z_{0} ,x_{0} ),\gamma (z_{0} ,x_{0} )^{'} )^{'}$; then, the local MLE of $\theta (.,.) = (\pi (.,.),\gamma (.,.)^{'} )^{'}$ is given by $\tilde{\pi }(z,x) = \pi^{*} (.,.)$ and $\tilde{\gamma }(z,x) = \gamma^{*} (.,.)$. Note that, however, since the $\pi (z)$ do not depend on $x$ and $\gamma (x)$ do not depend on $z$, the improved estimators of $\pi (z)$ and $\gamma (x)$ can be obtained using integrated backfitting approach. Thus, given the estimates $\tilde{\pi }(z,x)$ and $\tilde{\gamma }(z,x)$, the initial estimates of $\pi (z)$ and $\gamma (x)$ (up to additive constants) are given by

$$ \begin{aligned} \tilde{\pi }(z) = & \int {\tilde{\pi }(z,x)f_{X} (x){\text{d}}x} \\ \gamma (x) = & \int {\tilde{\gamma }(z,x)f_{Z} (z){\text{d}}z} , \\ \end{aligned} $$

where $f_{X} (x)$ and $f_{Z} (z)$ are marginal densities of $X$ and $Z$, respectively. Now given the initial estimator of $\tilde{\pi }(z)$, for every fixed set points $x_{0}$ within the closed support of $X$, the improved estimator of $\gamma (x_{0} )$ is defined as $\hat{\gamma }(x_{0} ) = \hat{\gamma }_{0} (x_{0} ) = \hat{\gamma }_{0}$ where $\hat{\gamma }_{0}$ is the first minimizer of the following plug-in conditional local log-likelihood function:

$$ L_{7n} \left( {\tilde{\pi }(z_{i} ),\gamma_{0} (x_{0} ),\Gamma_{1} (x_{0} )} \right) = \sum\limits_{i = 1}^{n} {\left\{ {\log f(Y_{i} ;\tilde{\pi }(z_{i} ),\gamma_{0} (x_{0} ) + \Gamma_{1} (x_{0} )(X_{i} - x_{0} ))} \right\}} K_{{H_{2} }} (X_{i} - x_{0} ). $$

(B.3)

Given the estimates of $\hat{\gamma }(x_{i} )$, we can obtain the improved estimator of $\pi (z_{i} )$, denote by $\hat{\pi }(z_{0} ) = \hat{\pi }_{0} (z_{0} ) = \hat{\pi }_{0}$ where $\hat{\pi }_{0}$ is the first maximizer the following plug-in conditional local log-likelihood function:

$$ L_{8n} \left( {\pi (z_{0} ),\hat{\gamma }(x_{i} )} \right) = \sum\limits_{i = 1}^{n} {\left\{ {\log f(Y_{i} ;\hat{\gamma }(x_{i} ),\pi_{0} (z_{0} ) +\Pi _{1} (z_{0} )(Z_{i} - z_{0} ))} \right\}} W_{{A_{2} }} (Z_{i} - z_{0} ). $$

(B.4)