Methodology

The SW(2023) Estimator

The estimator of Simar & Wilson (2023) extends nonparametric stochastic frontier analysis to technologies with multiple outputs. Given \(n\) observations \((X_i, Y_i)\) where \(X_i \in \mathbb{R}^p\) are inputs and \(Y_i \in \mathbb{R}^q\) are outputs, the model is:

\[U_i = \varphi(Z_i) + \|d\| \varepsilon_i, \quad \varepsilon_i = v_i - \eta_i\]

where \(v_i \sim N(0, \sigma_\varepsilon^2(Z_i))\) is noise and \(\eta_i \sim N^+(0, \sigma_\eta^2(Z_i))\) is inefficiency, both potentially heterogeneous in \(Z_i\).

Rotation Transform

A direction vector \(d \in \mathbb{R}^{p+q}\) is chosen (typically the sample mean or median of \((X, Y)\)). The data are rotated to scalar coordinates:

\[U_i = d^\top (X_i, Y_i) / \|d\|^2, \quad Z_i = \text{projection of } (X_i,Y_i) \text{ orthogonal to } d.\]

Conditional Moment Estimation (LLLS)

Three conditional moments of \(U\) given \(Z\) are estimated by local linear least squares (LLLS):

\[\begin{split}r_1(z) &= E[U \mid Z=z] \\ r_2(z) &= E[\varepsilon^2 \mid Z=z] \\ r_3(z) &= E[\varepsilon^3 \mid Z=z]\end{split}\]

where \(\varepsilon_i = U_i - \hat{r}_1(Z_i)\). Bandwidth is selected by leave-one-out cross-validation (LOO-CV) or Silverman’s rule.

Inefficiency Estimation

Two estimators for \(\sigma_\eta(z)\) are implemented:

SVKZ (Simar, Van Keilegom & Zelenyuk):

\[\hat\sigma_\eta(z) = \left(\frac{-\hat{r}_3(z)}{a_3^+}\right)^{1/3}\]

HMS (Hafner, Manner & Simar) — handles wrong skewness (\(\hat{r}_3 > 0\)) by setting \(\hat\sigma_\eta = 0\) at those observations:

\[\begin{split}\hat\sigma_\eta(z) = \begin{cases} \left(\dfrac{-\hat{r}_3(z)}{a_3^+}\right)^{1/3} & \hat{r}_3(z) \le 0 \\ 0 & \hat{r}_3(z) > 0 \end{cases}\end{split}\]

where \(a_3^+ = -\sqrt{2/\pi}(1 - 4/\pi)\).

JLMS Efficiency

Individual efficiency scores are obtained via the JLMS formula:

\[\hat E[\eta_i \mid \hat\varepsilon_i] = \mu_i^* + \sigma_i^* \frac{\phi(\mu_i^*/\sigma_i^*)}{\Phi(\mu_i^*/\sigma_i^*)}\]

giving efficiency index \(\exp(-\hat\eta_i) \in (0,1]\).

4-Component Panel Extension

For panel data with \(T\) time periods per firm, the model decomposes:

\[U_{it} = \varphi(Z_{it}) + \|d\| v_{it} - \|d\| u_{it} + \|d\| \alpha_i - \|d\| \mu_i\]

  • \(v_{it} \sim N(0,\sigma_v^2)\) — transient noise

  • \(u_{it} \sim N^+(0,\sigma_u^2)\) — transient inefficiency

  • \(\alpha_i \sim N(0,\sigma_\alpha^2)\) — individual heterogeneity

  • \(\mu_i \sim N^+(0,\sigma_\mu^2)\) — persistent inefficiency

Identification follows Colombi et al. (2014): within-individual variation identifies transient components, between-individual variation identifies persistent components.

References

  • Simar, L. & Wilson, P.W. (2023). Nonparametric, Stochastic Frontier Models with Multiple Inputs and Outputs. JBES, 41(4), 1391–1403.

  • Hafner, C.M., Manner, H. & Simar, L. (2018). The “wrong skewness” problem in stochastic frontier models. Econometric Reviews, 37(4), 380–400.

  • Colombi, R., Kumbhakar, S.C., Martini, G. & Vittadini, G. (2014). Closed-skew normality in stochastic frontiers with individual effects and long/short-run efficiency. Journal of Productivity Analysis, 42, 123–136.