A hierarchical panel data stochastic frontier model for the estimation of stochastic metafrontiers

Abstract

This paper proposes a stochastic frontier model with three composed errors, and therefore six error components. As in the metafrontier literature, firms belong to groups with a group-specific frontier. A firm has a level of short-run and long-run inefficiency relative to its group-specific frontier, as in existing models with two composed errors and four error components. But now there is also a group-specific inefficiency, that is, a shortfall of the group-specific frontier from the best practice metafrontier. The paper shows how to estimate this model and how to extract predictions of the various inefficiencies.

Appendix

We wish to derive an expression for the density of $\varepsilon_{\left( g \right)} = \left( {\varepsilon_{11} , \ldots ,\varepsilon_{1T} , \ldots ,\varepsilon_{{n_{g} 1}} , \ldots ,\varepsilon_{{n_{g} T}} } \right)^{{\prime }}$. The starting point will be the density of $\xi_{g} = \left( {c_{1} , \ldots ,c_{{n_{g} }} ,w_{g} ,u_{11} , \ldots ,u_{{n_{g} T}} } \right).$ Since we are dealing with a specific group, group g, we will simplify the notation, for this “Appendix” only, by omitting the subscript g. Thus we write $\varepsilon$ in place of $\varepsilon_{g}$, $\xi$ in place of $\xi_{g}$, $w$ in place of $w_{g}$ and n in place of $n_{g}$. We will use results on the closed skew-normal distribution from González-Farías et al. (2004b) (hereafter GDG).

A p-dimensional random variable Z is distributed as ${\text{CSN}}_{p,q} \left( {\mu ,\varSigma ,D,\nu ,\Delta } \right)$ if its density is $f\left( z \right) = C\varphi_{p} \left( {z;\mu ,\varSigma } \right)\varPhi_{q} \left( {D\left( {z - \mu } \right);\nu ,\Delta } \right)$. Here $\varphi_{p}$ and $\varPhi_{q}$ are the p-variate normal density and the q-variate normal cdf, respectively, and $C^{ - 1} = \varPhi_{q} \left( {0;\nu ,\Delta + D\varSigma D^{\prime}} \right)$. The dimensions of the parameters are as follows: $\mu :p \times 1, \varSigma :p \times p, D:q \times p,\nu :q \times 1,\Delta :q \times q$. The relevance of this to the our model is that the composed error c_i, with parameters $\lambda_{c}$ and $\sigma_{c}^{2}$, is distributed as ${\text{CSN}}_{1,1} \left( {0,\sigma_{c}^{2} , - \frac{{\lambda_{c} }}{{\sigma_{c} }},0,1} \right)$, and similarly for w and $u_{it}$.

Proposition 2.4.1 of GDG says that independent marginally CSN random variables are jointly CSN. Generically, if $Z = \left( {Z_{1}^{{\prime }} , \ldots ,Z_{k}^{{\prime }} } \right)^{{\prime }}$ where the $Z_{j}$ are mutually independent and $Z_{j}$ ~ ${\text{CSN}}_{{p_{j} ,q_{j} }} \left( {\mu_{j} ,\varSigma_{j} ,D_{j} ,\nu_{j} ,\Delta_{j} } \right)$, then Z ~ $CSN_{p,q} \left( {\mu ,\varSigma ,D,\nu ,\Delta } \right)$, where $p = \mathop \sum \limits_{j} p_{j}$, $q = \mathop \sum \limits_{j} q_{j}$, $\mu = \left( {\mu_{1} ', \ldots ,\mu_{k} '} \right)'$, $\nu = \left( {\nu_{1}^{{\prime }} , \ldots ,\nu_{k}^{{\prime }} } \right)^{{\prime }}$, $\varSigma = \oplus_{j = 1}^{k} \varSigma_{j}$, $D = \oplus_{j = 1}^{k} D_{j}$, $\Delta = \oplus_{j = 1}^{k} \Delta_{j}$. Here ⊕ is the matrix direct sum operator that makes matrices A and B into a block diagonal matrix: $A \oplus B = \left[ {\begin{array}{*{20}c} A & O \\ O & B \\ \end{array} } \right]$. In our case, this implies that $\xi \sim\,{\text{CSN}}_{q,q} \left( {\mu ,\varSigma ,D,\nu ,\Delta } \right)$ where $q = n\left( {T + 1} \right) + 1$ and:

$$ \begin{aligned} \mu & = 0,\nu = 0\;({\text{both}}\;q \times 1) \\ \varSigma & = \sigma_{c}^{2} I_{n} \oplus \sigma_{w}^{2} \oplus \sigma_{u}^{2} I_{nT} \quad ({\text{a}}\;{\text{diagonal}}\;{\text{matrix}}\;{\text{of}}\;{\text{dimension}}\;q) \\ D & = - \frac{{\lambda_{c} }}{{\sigma_{c} }}I_{n} \oplus - \frac{{\lambda_{w} }}{{\sigma_{w} }} \oplus - \frac{{\lambda_{u} }}{{\sigma_{u} }}I_{nT} \quad ({\text{a}}\;{\text{diagonal}}\;{\text{matrix}}\;{\text{of}}\;{\text{dimension}}\;q) \\ \Delta & = I_{q} \\ \end{aligned} $$

Proposition 2.3.1 of GDG says that linear combinations of jointly CSN random variables are jointly CSN. Generically, suppose that Z ~ ${\text{CSN}}_{p,q} \left( {\mu ,\varSigma ,D,\nu ,\Delta } \right)$ and let A be $m \times p,m \le p,$ rank(A) = m. Then AZ ~ ${\text{CSN}}_{m,q} \left( {\mu_{A} ,\varSigma_{A} ,D_{A} ,\nu ,\Delta_{A} } \right)$, where $\mu_{A} = A\mu$, $\varSigma_{A} = A\varSigma A'$, $D_{A} = D\varSigma A'\varSigma_{A}^{ - 1}$, $\Delta_{A} = \Delta + D\varSigma D^{{\prime }} - D\varSigma A^{{\prime }} \varSigma_{A}^{ - 1} A\varSigma D^{{\prime }}$. In our case, $\varepsilon = A\xi$ where A is $nT \times q$ and is defined as follows:

$$ \begin{aligned}& A = \left[ {B_{1} ,B_{2} ,B_{3} } \right] \\ &B_{1} = I_{n} \otimes 1_{T} ,\;{\text{where}}\;1_{T} \;{\text{is}}\;{\text{a}}\;T \times 1\;{\text{vector}}\;{\text{of}}\;{\text{ones}}({\text{so}}\;B_{1} \;{\text{is}}\;{\text{of}}\;{\text{dimension}}\;nT \times n) \\ &B_{2} = 1_{nT} \;(nT \times 1) \\ &B_{3} = I_{nT} \;(nT \times nT) \\ \end{aligned} $$

Therefore, $\varepsilon \sim CSN_{nT,q} \left( {\mu_{A} ,\varSigma_{A} ,D_{A} ,\nu ,\Delta_{A} } \right)$ and the density of $\varepsilon$ is

$$ f\left( \varepsilon \right) = C_{A} \varphi_{nT} \left( {\varepsilon ;\mu_{A} ,\varSigma_{A} } \right)\varPhi_{q} \left( {D_{A} \left( {\varepsilon - \mu_{A} } \right);\nu ,\Delta_{A} } \right) $$

(A1)

where $C_{A}^{ - 1} = \varPhi_{q} \left( {0;\nu ,\Delta_{A} + D_{A} \varSigma_{A} D_{A}^{{\prime }} } \right).$

Some of these arguments of the density can be simplified. For example, $\mu_{A} = 0$, $\nu = 0$, and $\varSigma_{A} = \sigma_{u}^{2} I_{nT} + \sigma_{c}^{2} \left( {I_{n} \otimes 1_{T} 1_{T}^{'} } \right) + \sigma_{w}^{2} 1_{nT} 1_{nT} '$. However, $D_{A}$ and $\Delta_{A}$ are rather complicated, and, importantly, $\Delta_{A}$ does not have any special algebraic structure (e.g., block diagonal) that would allow the dimensionality of the integral implicit in the cdf $\varPhi_{q} \left( {D_{A} \varepsilon ;0,\Delta_{A} } \right)$ to be reduced. So we are left with the task of evaluating the joint cdf of a multivariate normal of dimension $q = n\left( {T + 1} \right) + 1$. This is not likely to be practical.