Maximum likelihood estimation of the stochastic frontier model with endogenous switching or sample selection

Abstract

Heckman’s (Ann Econ Soc Meas 15:475–492, 1976; Econometrica 47(1):153–161, 1979) sample selection model has been employed in many applications of linear or nonlinear regression studies. It is well known that ignoring the sample selectivity may result in estimation bias of the estimator. Although the stochastic frontier (SF) model with sample selection has been investigated in Greene (J Product Anal 34:15–24, 2010), we intend to extend the model in several directions in this paper. First, we extend the distribution of the inefficiency from the half normal to truncated normal distribution. Second, we discuss the likelihood estimation method for the SF model with sample selection and also its most common incarnation, endogenous switching. Third, we suggest a simple framework to derive the closed form of the likelihood function using the closed skew-normal distribution. Fourth, we propose the estimator for the technical efficiency index due to Battese and Coelli (Empir Econ 20(2):325–332, 1995) based on the sample selection information. Finally, we also demonstrate the approach using the Taiwan hotel industry data.

Appendices

Appendix 1

We use the symbol “$\oplus$” to indicate the matrix direct sum operator: for any two matrices A and B, $A \oplus B = \left( {\begin{array}{*{20}c} A & O \\ O & B \\ \end{array} } \right)$. Property 1 suggests that the joint distribution of the independent CSN random vectors is still a CSN distribution.

Property 1

(Proposition 2.4.1 of GDG (2004)) If $y_{1} ,{ \ldots },y_{n}$ are independent random vectors with $y_{i} \sim CSN_{{p_{i} ,q_{i} }} \left( {\pi_{i} ,\varSigma_{i} ,\varGamma_{i} ,\kappa_{i} ,\Delta_{i} } \right)$ , the joint distribution of $y_{1} ,{ \ldots },y_{n}$ is

$$y = \left( {y_{1}^{\text{T}} ,{ \ldots }y_{n}^{\text{T}} } \right)^{\text{T}} \sim CSN_{{p^{*} ,q^{*} }} \left( {\pi^{*} ,\varSigma^{*} ,\varGamma^{*} ,\kappa^{*} ,\Delta^{*} } \right)$$

(27)

where $p^{*} = \sum\nolimits_{i = 1}^{n} {p_{i} }$, $q^{*} = \sum\nolimits_{i = 1}^{n} {q_{i} }$, $\pi^{*} = \left( {\pi_{1}^{\text{T}} ,{ \ldots },\pi_{n}^{\text{T}} } \right)^{\text{T}}$, $\varSigma^{*} = \mathop \oplus \limits_{i = 1}^{n} \, \varSigma_{i}$, $\varGamma^{*} = \mathop \oplus \limits_{i = 1}^{n} \, \varGamma_{i}$, $\kappa^{*} = \left( {\kappa_{1}^{\text{T}} ,{ \ldots },\kappa_{n}^{\text{T}} } \right)^{\text{T}}$ , and $\Delta^{*} = \mathop \oplus \limits_{i = 1}^{n} \, \Delta_{i}$.

Property 2 suggests that a linear transformation of a multivariate CSN random vector is still CSN distributed.

Property 2

(Proposition 2.3.1 of GDG (2004)) If $y\sim CSN_{p,q} \left( {\pi ,\varSigma ,\varGamma ,\kappa ,\Delta } \right)$ and let A be a $n \times p$ matrix of rank n, where $n \le p$ . Then

$$Ay\sim CSN_{n,q} \left( {\pi_{A} ,\varSigma_{A} ,\varGamma_{A} ,\kappa ,\Delta_{A} } \right)$$

(28)

where $\pi_{A} = A\pi$, $\varSigma_{A} = A\varSigma A^{\text{T}}$, $\varGamma_{A} = \varGamma \varSigma A^{\text{T}} \varSigma_{A}^{ - 1}$ , and $\Delta_{A} = \Delta + \varGamma \varSigma \varGamma^{\text{T}} - \varGamma \varSigma A^{\text{T}} \varSigma_{A}^{ - 1} A\varSigma \varGamma^{\text{T}}$.

The following Property 3 states the conditional distribution of the CSN.

Property 3

(Proposition 2.3.2 of GDG (2004)) If $y \sim CSN_{p,q} \left( {\pi ,\varSigma ,\varGamma ,\kappa ,\Delta } \right)$ then for two subvectors $y_{1}$ and $y_{2}$, where $y^{\text{T}} = \left( {\begin{array}{*{20}c} {y_{1}^{\text{T}} } & {y_{2}^{\text{T}} } \\ \end{array} } \right)$, $y_{1}$ is k-dimensional, $1 \le k \le p$ , and $\pi$, $\varSigma$, and $\varGamma$ are partitioned as $\pi = \left( {\begin{array}{*{20}c} {\pi_{1} } \\ {\pi_{2} } \\ \end{array} } \right) , { }\varSigma { = }\left( {\begin{array}{*{20}c} {\varSigma_{11} } & {\varSigma_{12} } \\ {\varSigma_{21} } & {\varSigma_{22} } \\ \end{array} } \right), \varGamma = \left( {\begin{array}{*{20}c} {\varGamma_{1} } & {\varGamma_{2} } \\ \end{array} } \right),$ then the conditional distribution of $y_{2}$ given $y_{1} = y_{10}$ is

$$CSN_{p - k,q} \left( {\pi_{2} + \varSigma_{21} \varSigma_{11}^{ - 1} \left( {y_{10} - \pi_{1} } \right),\varSigma_{22 \cdot 1} ,\varGamma_{2} ,\kappa - \varGamma^{*} (y_{10} - \pi_{1} ),\Delta } \right),$$

where $\varGamma^{*} = \varGamma_{1} + \varGamma_{2} \varSigma_{21} \varSigma_{11}^{ - 1}$, $\Delta^{*} = \Delta + \varGamma_{2} \varSigma_{22 \cdot 1} \varGamma_{2}^{\text{T}}$, and $\varSigma_{22 \cdot 1} = \varSigma_{22} - \varSigma_{21} \varSigma_{11}^{ - 1} \varSigma_{12}$.

Appendix 2

Lemma B.1

Under assumption [A1], both $v_{i} |\{ e_{i} > - w_{i}^{\text{T}} \gamma \}$ and $v_{i} |\{ e_{i} \le - w_{i}^{\text{T}} \gamma \}$ follow CSN distributions. More specifically,

$$v_{i} |d_{i} \sim CSN_{1,1} \left( {0,\sigma_{v}^{2} ,(2d_{i} - 1)\frac{\rho }{{\sigma_{v} }}, - (2d_{i} - 1)w_{i}^{\text{T}} \gamma ,1 - \rho^{2} } \right) .$$

(29)

Proof of Lemma B.1

Using the result of Theorem 2.1.1 of Tong (1990), we have $e|v\sim N (\rho v/\sigma_{v} , { }1 - \rho^{2} )$ under assumption [A1]. Let r be a known constant and $\varPhi_{z} ( \cdot )$ be the cdf of a standard normal random variable, then

• (i)
  $$\begin{aligned} f_{v|e} (v|e > r) & = & \frac{{f_{v} (v)}}{\Pr (e > r)}\Pr (e > r|v) = \frac{{\phi_{1} (v;0,\sigma_{v}^{2} )}}{{\varPhi_{z} ( - r)}}\left[ {1 - \varPhi_{1} \left( {r;\frac{\rho }{{\sigma_{v} }}v,1 - \rho^{2} } \right)} \right] \\ & & = & \frac{{\phi_{1} (v;0,\sigma_{v}^{2} )}}{{\varPhi_{z} ( - r)}}\varPhi_{1} \left( {\frac{\rho }{{\sigma_{v} }}v;r,1 - \rho^{2} } \right){\text{ is }}CSN_{1,1} \left( {0,\sigma_{v}^{2} ,\frac{\rho }{{\sigma_{v} }},r,1 - \rho^{2} } \right). \\ \end{aligned}$$
• (ii)
  $$\begin{aligned} f_{v|e} (v|e \le r) = & \frac{{f_{v} (v)}}{\Pr (e \le r)}\Pr (e \le r|v) = \frac{{\phi_{1} (v;0,\sigma_{v}^{2} )}}{{\varPhi_{z} (r)}}\varPhi_{1} \left( {r;\frac{\rho }{{\sigma_{v} }}v,1 - \rho^{2} } \right) \\ = \frac{{\phi_{1} (v;0,\sigma_{v}^{2} )}}{{\varPhi_{z} (r)}}\varPhi_{1} \left( { - \frac{\rho }{{\sigma_{v} }}v; - r,1 - \rho^{2} } \right){\text{ is }}CSN_{1,1} \left( {0,\sigma_{v}^{2} , - \frac{\rho }{{\sigma_{v} }}, - r,1 - \rho^{2} } \right). \\ \end{aligned}$$

Denote $d = 1(e > r)$, then (i) and (ii) together implies that $f_{v|e} (v|d)$ is $CSN_{1,1} \left( {0,\sigma_{v}^{2} ,(2d - 1)\frac{\rho }{{\sigma_{v} }},(2d - 1)r,1 - \rho^{2} } \right)$. □

Lemma B.2

Under assumption [A2], $u_{i} \sim CSN_{1,1} (\mu_{i} ,\sigma_{u}^{2} ,1, - \mu {}_{i},0)$.

Proof Lemma B.2

By Lemma 13.6.1 of Dominguez-Molina, González-Farías and Ramos-Quiroga (2004), the moment generating function of the truncated normal distribution is

$$M_{u} (t) = \frac{{\varPhi_{z} \left( {\tfrac{{\mu_{i} + t\sigma_{u}^{2} }}{{\sigma_{u} }}} \right)}}{{\varPhi_{z} \left( {\tfrac{{\mu_{i} }}{{\sigma_{u} }}} \right)}}e^{{t\mu + \tfrac{1}{2}t^{2} \sigma_{u}^{2} }} = \frac{{\varPhi \left( {t\sigma_{u}^{2} ; - \mu_{i} ,\sigma_{u}^{2} } \right)}}{{\varPhi \left( {0; - \mu_{i} ,\sigma_{u}^{2} } \right)}}e^{{t\mu_{i} + \frac{1}{2}t^{2} \sigma_{u}^{2} }} .$$

(30)

By comparing (30) and (6). It follows that $u_{i}$ actually follows the $CSN_{1,1} (\mu_{i} ,\sigma_{u}^{2} ,1, - \mu_{i} ,0)$ distribution. □

Under [A2], $v_{i} |d_{i}$ and $u_{i} |d_{i}$ are independent to each other. Then it follows from Lemma B.1, Lemma B.2 and Property 1 that

$$\left( {v_{i} ,u_{i} } \right)^{\text{T}} |d_{i} \sim CSN_{2,2} \left( {\pi_{i}^{*} ,\varSigma^{*} ,\varGamma_{i}^{*} ,\kappa_{i}^{*} ,\Delta^{*} } \right) ,$$

(31)

where $\pi_{i}^{*} = \left( {\begin{array}{*{20}c} 0 \\ {\mu_{i} } \\ \end{array} } \right),$ $\varSigma^{*} = \left( {\begin{array}{*{20}c} {\sigma_{v}^{2} } & 0 \\ 0 & {\sigma_{u}^{2} } \\ \end{array} } \right),$ $\varGamma_{i}^{*} = \left( {\begin{array}{*{20}c} {(2d_{i} - 1)\rho /\sigma_{v} } & 0 \\ 0 & 1 \\ \end{array} } \right),$ $\kappa_{i}^{*} = \left( {\begin{array}{*{20}c} { - (2d_{i} - 1)w_{i}^{\text{T}} \gamma } \\ { - \mu_{i} } \\ \end{array} } \right),$ $\Delta^{*} = \left( {\begin{array}{*{20}c} {1 - \rho^{2} } & 0 \\ 0 & 0 \\ \end{array} } \right).$ Since $\varepsilon_{i} |d_{i}$ can be represented a linear combination of $v_{i} |d_{i}$ and $u_{i} |d_{i}$ and Property 2 suggests that the linear combination of $v_{i} |d_{i}$ and $u_{i} |d_{i}$ is still CSN, we can easily obtain the conditional probability density function of the model in (3). The main result is summarized in Theorem 1.

Proof of Theorem 1

Let $A = (1, - 1)$. Given the result in (31), $\left( {v_{i} ,u_{i} } \right)^{\text{T}} |d_{i} \sim CSN_{2,2} \left( {\pi_{i}^{*} ,\varSigma^{*} ,\varGamma_{i}^{*} ,\kappa_{i}^{*} ,\Delta^{*} } \right)$, and $\varepsilon_{i} = A(v_{i} ,u_{i} )^{\text{T}}$, then it follows from Property 2 that $\varepsilon_{i} |d_{i} \sim CSN_{1,2} \left( {\pi_{A,i} ,\varSigma_{A} ,\varGamma_{A,i} ,\kappa_{A,i} ,\Delta_{A,i} } \right)$, where $\pi_{A,i} = A\pi_{i}^{*} = (1, - 1)\left( \begin{gathered} 0 \hfill \\ \mu_{i} \hfill \\ \end{gathered} \right) = - \mu_{i}$, $\varSigma_{A} = A\varSigma^{*} A^{\text{T}} = (\sigma_{v}^{2} + \sigma_{u}^{2} ),$ $\varGamma_{A,i} = \varGamma_{i}^{*} \varSigma^{*} A^{\text{T}} \varSigma_{A}^{ - 1} = \frac{1}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}\left( \begin{gathered} \left( {2d - 1} \right)\rho \sigma_{v} \\ - \sigma_{u}^{2} \\ \end{gathered} \right),\;\kappa_{A,i} = \kappa_{i}^{*} \;{\text{and }}\Delta_{A,i} = \Delta^{*} + \varGamma_{i}^{*} \varSigma^{*} \varGamma_{i}^{{*{\text{T}}}} - \varGamma_{i}^{*} \varSigma^{*} A^{T} \varSigma_{A}^{ - 1} A\varSigma^{*} \varGamma_{i}^{{*{\text{T}}}}$ $= \frac{1}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}\left( {\begin{array}{*{20}c} {(1 - \rho^{2} )\sigma_{v}^{2} + \sigma_{u}^{2} } & {\left( {2d_{i} - 1} \right)\rho \sigma_{v} \sigma_{u}^{2} } \\ {\left( {2d_{i} - 1} \right)\rho \sigma_{v} \sigma_{u}^{2} } & {\sigma_{v}^{2} \sigma_{u}^{2} } \\ \end{array} } \right).$

To obtain the corresponding pdf of $CSN_{1,2} \left( {\pi_{A,i} ,\varSigma_{A} ,\varGamma_{A,i} ,\upkappa_{A,i} ,\Delta_{A,i} } \right)$, as defined in (5). (i) Let $\varPi = \Delta_{A,i} + \varGamma_{A,i} \varSigma_{A} \varGamma_{A,i}^{\text{T}}$ $= \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & {\sigma_{u}^{2} } \\ \end{array} } \right)$. Since $\varPi$ is a diagonal matrix, it follows that $C^{ - 1} = \varPhi_{2} \left( {0;\kappa_{A,i} ,\varPi } \right) = \varPhi_{1} \left( {0; - \left( {2d_{i} - 1} \right)w_{i}^{\text{T}} \gamma ,1} \right)\varPhi_{1} \left( {0; - \mu_{i} ,\sigma_{u}^{2} } \right) = \varPhi_{z} \left( {(2d_{i} - 1)w_{i}^{\text{T}} \gamma } \right)\varPhi_{z} \left( {\mu_{i} /\sigma_{u} } \right).$ (ii) $\phi_{1} \left( {\varepsilon_{i} ; - \mu_{i} ,\sigma_{v}^{2} + \sigma_{u}^{2} } \right) = \frac{1}{{\sqrt {\sigma_{v}^{2} + \sigma_{u}^{2} } }}\phi_{z} \left( {\frac{{\varepsilon_{i} + \mu_{i} }}{{\sqrt {\sigma_{v}^{2} + \sigma_{u}^{2} } }}} \right).$ (iii) $\varPhi_{2} \left( {\varGamma_{A} (\varepsilon_{i} - \mu_{i} );\kappa_{i}^{*} ,\Delta_{A,i} } \right) =\varPhi_{2} \left( 0; \kappa_{A,i}-\Gamma_{A,i}(\varepsilon_i-\mu_{i}), \Delta_{A,i}\right)$. Therefore, (i–iii) together imply (8). □

Proof of Corollary 2

If $\rho = 0$, then $\Delta_{A} = \left( {\begin{array}{*{20}c} 1 & 0 \\ 0 & {\sigma_{v}^{2} \sigma_{u}^{2} /(\sigma_{v}^{2} + \sigma_{u}^{2} )} \\ \end{array} } \right)$, $\varPhi_{2} \left( {\varGamma_{A,i} (\varepsilon_{i} + \mu_{i} );\kappa_{A,i} ,\Delta_{A,i} } \right)$=$\varPhi_{z} \left( {(2d_{i} - 1)w_{i}^{\text{T}} \gamma } \right)$ $\times \varPhi_{1} \left( { - \frac{{\sigma_{u}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}(\varepsilon_{i} + \mu_{i} ); - \mu_{i} ,\frac{{\sigma_{v}^{2} \sigma_{u}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}} \right)$, where $\varPhi_{1} \left( { - \frac{{\sigma_{u}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}(\varepsilon_{i} + \mu_{i} ); - \mu_{i} ,\frac{{\sigma_{v}^{2} \sigma_{u}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}} \right) =$ $\varPhi_{z} \left( { - \frac{{\sigma_{u} /\sigma_{v} }}{{\sqrt {\sigma_{v}^{2} + \sigma_{u}^{2} } }}\varepsilon_{i} + \frac{{\sigma_{v} /\sigma_{u} }}{{\sqrt {\sigma_{v}^{2} + \sigma_{u}^{2} } }}\mu_{i} } \right)$. Substituting the result into (8), we obtain (9). □

Proof of Theorem 3

Let $B = \left( {\begin{array}{*{20}c} 1 & { - 1} \\ 0 & 1 \\ \end{array} } \right)$, then it follows from (5) and Property 2 that $\left( {\varepsilon_{i} ,u_{i} } \right)^{\text{T}} |d_{i} = B \cdot \left( {v_{i} ,u_{i} } \right)^{\text{T}} |d_{i}$ $\sim CSN_{2,2} (\pi_{B,i} ,\varSigma_{B} ,\varGamma_{B,i} ,\kappa_{B,i} ,\Delta_{B} )$, where $\pi_{B,i} = \left( {\begin{array}{*{20}c} { - \mu_{i} } \\ {\mu_{i} } \\ \end{array} } \right)$, $\varSigma_{B} = \left( {\begin{array}{*{20}c} {\sigma_{v}^{2} + \sigma_{u}^{2} } & { - \sigma_{u}^{2} } \\ { - \sigma_{u}^{2} } & {\sigma_{u}^{2} } \\ \end{array} } \right)$, $\varGamma_{B,i} = \left( {\begin{array}{*{20}c} {(2d_{i} - 1)\frac{\rho }{{\sigma_{v} }}} & {(2d_{i} - 1)\frac{\rho }{{\sigma_{v} }}} \\ 0 & 1 \\ \end{array} } \right)$, $\Delta_{B} = \left( {\begin{array}{*{20}c} {1 - \rho^{2} } & 0 \\ 0 & 0 \\ \end{array} } \right)$, and $\kappa_{B,i} = \left( {\begin{array}{*{20}c} { - (2d_{i} - 1)w_{i}^{\text{T}} \gamma } \\ { - \mu } \\ \end{array} } \right)$. Then Property 3 suggests that $\left. {u_{i} } \right|\left\{ {\left. {\varepsilon_{i} } \right|d_{i} } \right\}\sim CSN_{1,2} (\tilde{\pi }_{i} ,\sigma_{*}^{2} ,\tilde{\varGamma }_{i} ,\tilde{\kappa }_{i} ,\tilde{\Delta })$, where $\tilde{\pi }_{i} = \frac{{\sigma_{v}^{2} \mu_{i} - \sigma_{u}^{2} \varepsilon_{i} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}$, $\sigma_{*}^{2} = \frac{{\sigma_{v}^{2} \sigma_{u}^{2} }}{{\sigma_{v}^{2} + \sigma_{u}^{2} }}$, $\tilde{\varGamma }_{i} = \left( {\begin{array}{*{20}c} {(2d_{i} - 1)\frac{\rho }{{\sigma_{v} }}} \\ 1 \\ \end{array} } \right)$, $\tilde{\kappa }_{i} = \left( {\begin{array}{*{20}c} {\frac{{(2d_{i} - 1)\rho }}{{\sigma_{v} }}\left( {\frac{{\sigma_{v}^{2} (\varepsilon_{i} + \mu_{i} )}}{{\sigma_{v}^{2} + \sigma_{u}^{2} }} - w_{i}^{\text{T}} \gamma } \right)} \\ {\frac{{\sigma_{u}^{2} (\varepsilon_{i} + \mu_{i} )}}{{\sigma_{v}^{2} + \sigma_{u}^{2} }} - \mu_{i} } \\ \end{array} } \right)$, and $\tilde{\Delta } = \Delta_{B} = \left( {\begin{array}{*{20}c} {1 - \rho^{2} } & 0 \\ 0 & 0 \\ \end{array} } \right)$. The corresponding moment generating function is $E(e^{{tu_{i} }} |\left. {\{ \varepsilon_{i} } \right|d_{i} \} ) = M_{{\left. u \right|\left. {\{ \varepsilon } \right|d\} }} (t) = \frac{{\varPhi_{2} (\ddot{\varGamma }_{i} t;\tilde{\kappa }_{i} ,\ddot{\Delta }_{i} )}}{{\varPhi_{2} (0;\tilde{\kappa }_{i} ,\ddot{\Delta }_{i} )}}e^{{t\tilde{\pi }_{i} + \frac{1}{2}t^{ 2} \sigma_{*}^{2} }}$, where $\ddot{\varGamma }_{i} = \tilde{\varGamma }_{i} \sigma_{*}^{2}$ and $\ddot{\Delta }_{i} = \tilde{\Delta } + \sigma_{*}^{2} \tilde{\varGamma }_{i} \tilde{\varGamma }_{i}^{\text{T}}$. Substituting $t = - 1$ gives the estimator of TE, $E(e^{{ - u_{i} }} |\left. {\{ \varepsilon_{i} } \right|d_{i} \} )$. □