Closed-skew normality in stochastic frontiers with individual effects and long/short-run efficiency

Abstract

This paper considers the estimation of Kumbhakar et al. (J Prod Anal. doi:10.1007/s11123-012-0303-1, 2012) (KLH) four random components stochastic frontier (SF) model using MLE techniques. We derive the log-likelihood function of the model using results from the closed-skew normal distribution. Our Monte Carlo analysis shows that MLE is more efficient and less biased than the multi-step KLH estimator. Moreover, we obtain closed-form expressions for the posterior expected values of the random effects, used to estimate short-run and long-run (in)efficiency as well as random-firm effects. The model is general enough to nest most of the currently used panel SF models; hence, its appropriateness can be tested. This is exemplified by analyzing empirical results from three different applications.

Appendix

Proof of Proposition 1

In the TTT SF model presented in Eq. (1), the random components b _i  − u _i0 − u _it  + e _it can be written as the sum of the time-independent terms (i.e., η _i  = b _i  − u _i0) and of the time-dependent terms (i.e., ɛ _it  = e _it  − u _it ). According to our assumptions, these two terms are independent in probability and are given by the difference of a normal-random variable and an independent, left-truncated-at-zero normal random variable. It is well known (Kumbhakar and Lovell 2000) that η _i has the following density:

$$f(\eta_i)=2\phi_1(\eta_i,0, \sigma_{b}^2+ \sigma_{1u}^2)\bar \Upphi_1\left(\frac{-\sigma_{1u}^2}{\sigma_{b}^2+ \sigma_{1u}^2}\eta_i,\sigma_{1u}^2 \left(1-\frac{\sigma_{1u}^2}{\sigma_{b}^2+ \sigma_{1u}^2}\right)\right).$$

Analogously, the density of ɛ _it (t = 1, 2, …, T) is:

$$f(\varepsilon_{it})=2\phi_1(\varepsilon_{it},0, \sigma_{e}^2+ \sigma_{2u,t}^2)\bar \Upphi_1\left(\frac{-\sigma_{2u,t}^2}{\sigma_{e}^2+ \sigma_{2u,t}^2}\varepsilon_{it},\sigma_{2u,t}^2\left(1-\frac{\sigma_{2u,t}^2}{\sigma_{e}^2+ \sigma_{2u,t}^2}\right)\right).$$

The previous two densities are (1,1) closed-skew normal densities and so the random components b _i  − u _i0 − u _it  + e _it of the vector ${\bf 1}_T b_i+\user2{Au}_i+\user2{e}_i=-\user2{A}(\eta_{i}, \varepsilon_{i1}, \dots, \varepsilon_{iT})^{'}$ are sums of two closed-skew normal random errors.

Let $\varvec{\Upupsilon}$ be the diagonal matrix with the ratios $\frac{\sigma_{2u,t}^2}{\sigma_{e}^2+\sigma_{2u,t}^2}$ on the main diagonal. From Theorem (3) of Gonzáles-Farías et al. (2004), it follows that the T + 1 independent-random variables η _i , ɛ _it (t = 1, 2, …, T) have a joint (T + 1,T + 1) closed-skew normal density function with parameters $\varvec{\nu}_0={\bf 0}, \varvec{\mu}_0= {\bf 0}$, and

$$\begin{aligned} \varvec{\Upgamma}_0=& \left[ \begin{array}{ll} \sigma_{1u}^2+\sigma_{b}^2& {\bf 0}^{\prime}_T \\ {\bf 0}_T& \varvec{\Uppsi}+\sigma_{e}^2 \user2{I}_T \end{array}\right], \user2{D}_0=\left[ \begin{array}{ll} \frac{-\sigma_{1u}^2}{\sigma_{b}^2+\sigma_{1u}^2}& {\bf 0}^{\prime}_T\\ {\bf 0}_T&-\varvec{\Upupsilon} \end{array}\right],\\ &\varvec{\Updelta}_0=\left[ \begin{array}{ll} \sigma_{1u}^2(1-\frac{\sigma_{1u}^2}{\sigma_{b}^2+ \sigma_{1u}^2})& {\bf 0}^{\prime}_T\\ {\bf 0}_T& \varvec{\Uppsi}(\user2{I}-\varvec{\Upupsilon}) \end{array}\right]. \end{aligned}$$

Finally, from the previous result and from Theorem (1) of Gonzáles-Farías et al. (2004), it follows that the T dimensional-random vector ${\bf 1}_T b_i+ \user2{Au}_i +\user2{e}_i=-\user2{A}(\eta_{i}, \varepsilon_{i1}, \dots, \varepsilon_{iT})^{'}$ with components $b_{i}-u_{i0}-u_{it}+e_{it}=\eta_{i}+\varepsilon_{it}$ has a (T,T + 1) closed-skew normal distribution with parameters $\varvec{\nu}={\bf 0}, \varvec{\mu}= {\bf 0}, \varvec{\Upgamma}= \user2{A}\varvec{\Upgamma}_0 \user2{A}^{\prime} =\varvec{\Upsigma} +\user2{AV}\,\user2{A}^{\prime}, \user2{D}=-\user2{D}_0 \varvec{\Upgamma}_0 \user2{A}^{\prime} \varvec{\Upgamma}^{-1}= \user2{R}, \varvec{\Updelta}=\varvec{\Updelta}_0+\user2{D}_0 \varvec{\Upgamma}_0\user2{D}^{\prime}_0-\user2{D}_0 \varvec{\Upgamma}_0\user2{A}^{\prime} \varvec{\Upgamma}^{-1} \user2{A} \varvec{\Upgamma}_0 \user2{D}^{\prime}_0= \varvec{\Uplambda}$. Because $\varvec{\mu}$ is a location parameter and $\user2{y}_i= {\bf 1}_T \beta_0+ \user2{X}_i\varvec{\beta} +{\bf 1}_T b_i+\user2{Au}_i +\user2{e}_i$, the statement of the Proposition follows.

Proof of Proposition 3

To prove b) note that:

$$\begin{aligned} &f(\user2{u}_i|\user2{y}_i)=\frac{f(\user2{y}_i|\user2{u}_i)f(\user2{u}_i)}{f(\user2{y}_i)}= \frac{\phi_T(\user2{y}_i, {\bf 1}_T \beta_0+ \user2{X}_i \varvec{\beta}+ \user2{Au}_i,\varvec{\Upsigma}) \phi_{T+1}(\user2{u}_i, \user2{V})2^{T+1}}{f(\user2{y}_i)} \\ &=\frac{\phi_T(\user2{y}_i, {\bf 1}_T \beta_0+ \user2{X}_i \varvec{\beta,\Upsigma} + \user2{AV}\, \user2{A}^{\prime}) \phi_{T+1}(\user2{u}_i, \user2{Rr}_i, \varvec{\Uplambda})2^q}{f(\user2{y}_i)}= \frac{\phi_{T+1}(\user2{u}_i, \user2{Rr}_i, \varvec{\Uplambda})}{\overline{\Upphi}_{T+1}(\user2{Rr}_i, \varvec{\Uplambda})} \end{aligned}$$

where the third equality follows from Lemma 2 of Arellano et al. (2005). Noting that $f(\user2{u}_i|\user2{y}_i)$ is the density of a left-truncated-at-zero multi-normal random variable, d) follows from Lemma 13.6.1 of Dominguez-Molina et al. (2004). To prove a) we observe that:

$$\begin{aligned} f(b_i|\user2{y}_i)=&\phi(b_i,\sigma^2_{b}{\bf 1}^{\prime}\varvec{\Updelta} \user2{r}_i, \tilde{\sigma}^2_{b})\\ \times &\frac{ \int_0^\infty\ldots \int_0^\infty \phi_q(u,\user2{Rr}_i- \user2{R} {\bf 1}_T \sigma^2_{b} \tilde{\sigma}^{-2}_{b}\tilde{b}_i,{\tilde{\varvec{\Uplambda}}}) du_0\ldots du_T}{\int_0^\infty\ldots \int_0^\infty \phi_q(u,R r_i, \Uplambda)du_0\ldots du_T}. \end{aligned}$$

(9)

Point a) of the proposition follows immediately. Observing that $f(b_i|\user2{y}_i)$ is a closed-skew normal density, point c) follows from the result shown in Eq. (3) on the moment-generating function of a closed-skew normal random variable.