ssfa function

Spatial stochastic frontier estimation

This function estimates the Spatial Stochastic Frontier model introduced by Fusco and Vidoli (2013) in the following form: [REMOVE_ME]$log(y_{i}) = log(f(x_{i};\beta_i)) +v_{i}-u_{i} [REMOVE_ME_2]$

[REMOVE_ME]$u_{i}=\rho \sum_{i}w_{i.}u_{i} + \widetilde{u_{i}} [REMOVE_ME_2]$

where $y_i$ are the outputs, $x_i$ the inputs, $v_i$ the stochastic noise, $u_{i}$ the inefficiency term, $rho$ the spatial lag, $w_{i.}$ a standardized row of the spatial weights matrix and $\widetilde{u_{i}}$ the stochastic noise of the inefficiency term.

Description

This function estimates the Spatial Stochastic Frontier model introduced by Fusco and Vidoli (2013) in the following form:

$$log(y_{i}) = log(f(x_{i};\beta_i)) +v_{i}-u_{i}$$ $$u_{i}=\rho \sum_{i}w_{i.}u_{i} + \widetilde{u_{i}}$$

where $y_i$ are the outputs, $x_i$ the inputs, $v_i$ the stochastic noise, $u_{i}$ the inefficiency term, $rho$ the spatial lag, $w_{i.}$ a standardized row of the spatial weights matrix and $\widetilde{u_{i}}$ the stochastic noise of the inefficiency term.

ssfa(formula, data = NULL, data_w = NULL, intercept = TRUE, pars = NULL, par_rho = TRUE, 
                 form = "cost")

Arguments

• formula: an object of class formula (or one that can be coerced to that class): a symbolic description of the model to be fitted.
• data: an optional data frame containing the variables in the model.
• data_w: a data frame containing the spatial weight matrix.
• intercept: logical. If true the model includes intercept.
• pars: initial values for the parameters to be estimated.
• par_rho: logical. If true the function estimates the Spatial Stochastic Frontier (SSFA) otherwise the classical Stochastic Frontier (SFA).
• form: specifies the form of the frontier model as "cost" or "production".

Returns

ssfa returns the following objects of class ssfa: - y: the dependent variable.

• x: the covariates.

• X: the model matrix.

• coef: the estimated coefficients.

• sc: the form of the frontier model estimated (-1 = cost, 1 = production).

• hess: a symmetric matrix giving an estimate of the Hessian at the solution found.

• logLik: the value of the log likelihood function.

• ols: the linear model for the LR-test.

• sigmau2: the estimation of sigmau2 (only if par_rho=FALSE): value of inefficiency variance.

• sigmau2_dmu: the estimation of sigmau2_dmu (only if par_rho=TRUE): value of the part of the inefficiency variance due to DMU's specificities.

• sigmau2_sar: the estimation of sigmau2_sar: value of the part of the inefficiency variance due to the spatial correlation.

• sigmav2: the estimation of sigmav2: value of the stochastic error variance.

• sigma2: the estimation of sigma2: value of the total variance.

• rho: the estimation of the spatial lag parameter rho.

• fun: the distribution of the inefficiency term u.

• list_w: a listw object from nb2listw (See nb2listw ).

Note

NOTE 1: In this version the distribution of the inefficiency term $u$ is only "half-normal".

NOTE 2: The method used to maximize the log likelihood function is the Newton-Raphson. Please see the R function maxNR of the maxLik package for details (Henningsen and Toomet (2011)).

NOTE 3: Please note that the classical SFA inefficiency variance sigmau2, in the SSFA, is decomposed into sigmau2_dmu and sigmau2_sar, respectively the part of inefficiency variance due to DMU's specificities and to the spatial dependence, i.e. sigmau2 = sigmau2_dmu + sigmau2_sar and consequently the total variance is given by sigma2 = sigmau2_dmu + sigmau2_sar + sigmav2.

References

Battese, G. E., and T. J. Coelli (1995). A Model for Technical Inefficiency Effects in a Stochastic Frontier Production Function for Panel Data. Empirical Economics 20(2): 325-332.

Fusco, E. and Vidoli, F. (2013). Spatial stochastic frontier models: controlling spatial global and local heterogeneity, International Review of Applied Economics, 27(5) 679-694.

Fusco, E. (2020). Spatial Dependence in Efficiency Parametric Models: A Generalization and Simulation Studies, "Scienze Regionali, Italian Journal of Regional Science" Speciale/2021, 595-618.

Kumbhakar, S. C., and C. A. K. Lovell (2000). Stochastic Frontier Analysis, Cambridge University Press.

Henningsen, A. and Toomet, O. (2011). maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3), 443-458.

Author(s)

Fusco E. and Vidoli F.

Examples

library(ssfa) 
data(SSFA_example_data)
data(Italian_W)
ssfa <- ssfa(log_y ~ log_x, data = SSFA_example_data, 
             data_w=Italian_W, form = "production", par_rho=TRUE)

### SSFA total variance decomposition 
sigma2 = ssfa$sigmau2_dmu + ssfa$sigmau2_sar + ssfa$sigmav2
sigma2
ssfa$sigma2