Package 'ssfa'

Title: Spatial Stochastic Frontier Analysis
Description: Spatial Stochastic Frontier Analysis (SSFA) is an original method for controlling the spatial heterogeneity in Stochastic Frontier Analysis (SFA) models, for cross-sectional data, by splitting the inefficiency term into three terms: the first one related to spatial peculiarities of the territory in which each single unit operates, the second one related to the specific production features and the third one representing the error term.
Authors: Elisa Fusco, Francesco Vidoli
Maintainer: Elisa Fusco <[email protected]>
License: GPL-3
Version: 1.2.2
Built: 2024-12-24 06:55:50 UTC
Source: CRAN

Help Index

Spatial Stochastic Frontier models

Description

The package implements the Spatial Stochastic Frontier model for cross-sectional data introduced by Fusco and Vidoli (2013). The method controls spatial heterogeneity in SFA models by splitting the inefficiency term into three parts: the first one related to spatial peculiarities of the territory in which each single unit operates, the second one related to the specific production features and the third one representing the error term.

Author(s)

Elisa Fusco, Francesco Vidoli

Maintainer: Elisa Fusco <[email protected]>

References

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.

SSFA efficiency

Description

This function returns the technical efficiency of each producer (without local spatial effects) calculated by the Battese and Coelli (1988) formulation modified by using an autoregressive specification in the inefficiency term uu.

Usage

eff.ssfa(object, ...)

Arguments

object

an object of class ssfa.
...

further arguments for methods.

Value

Technical efficiency of each producer (without local spatial effects).

References

Battese, G. E., and T. J. Coelli (1988). Prediction of Firm-level Technical Efficiencies with a Generalized Frontier Production Function and Panel Data. Journal of Econometrics 38(3): 387-399.

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.

See Also

u.ssfa

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)
eff <- eff.ssfa(ssfa)

SSFA fitted values

Description

This function returns the fitted values of the original data used to estimate the SSFA model.

Usage

## S3 method for class 'ssfa'
fitted(object, ...)

Arguments

object

an object of class ssfa.
...

further arguments for methods.

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)
fitted.ssfa(ssfa)

Italian provinces spatial weights matrix example

Description

This is an example dataset that contains the 107 Italian provinces contiguity matrix (year 2008).

Usage

data(Italian_W)

Format

A data frame with 107 x 107 row-standardized distances between observations (Italian provinces).

References

http://www.istat.it/it/archivio/104317#confini.

Examples

data(Italian_W)

SFA half-normal log likelihood function

Description

This function is used to estimate the parameters of the classical SFA model where half-normal distribution of inefficiency term is assumed.

Usage

L_hNV(p, y = y, X = X, sc = sc)

Arguments

p

a vector with the parameters to be estimated.
y

the dependent variable.
X

the model matrix.
sc

specifies the form of the frontier model (-1 = cost, 1 = production).

Value

Value of the SFA log likelihood function.

SSFA half-normal log likelihood function

Description

This function is used to estimate the parameters of the SSFA model where half-normal distribution of inefficiency term is assumed.

Usage

L_hNV_rho(p, y = y, X = X, sc = sc, w = w, sigmau2_sar = sigmau2_sar)

Arguments

p

a vector with the parameters to be estimated.
y

the dependent variable.
X

the model matrix.
sc

specifies the form of the frontier model (-1 = cost, 1 = production).
w

the spatial weight matrix.
sigmau2_sar

is the variance of the spatial correlated part of the inefficiency term estimated into ssfa.fit function.

Value

Value of the SSFA log likelihood function.

Note

Please note that sigmau2_sar is not a free parameter because it is estimated into the ssfa.fit function.

See Also

ssfa

SSFA plot

Description

This function allows to plot the data and the fitted values obtained by SSFA model.

Usage

plot_fitted(x, y, object, xlab, ylab, main, ...)

Arguments

x

the x coordinates of points in the plot.
y

the y coordinates of points in the plot.
object

an object of class ssfa.
xlab

a title for the x axis.
ylab

a title for the y axis.
main

an overall title for the plot.
...

arguments to be passed to methods, such as graphical parameters (see par).

See Also

plot

Examples

library(ssfa) 
data(SSFA_example_data)
data(Italian_W)

#### SFA and SSFA comparison 
sfa <- ssfa(log_y ~ log_x, data = SSFA_example_data, data_w=Italian_W, 
             form = "production", par_rho=FALSE)
ssfa <- ssfa(log_y ~ log_x, data = SSFA_example_data, data_w=Italian_W, 
 form = "production", par_rho=TRUE)

sfa_fitted <- fitted.ssfa(sfa)
plot_fitted(SSFA_example_data$log_x, SSFA_example_data$log_y, ssfa)
lines(sort(SSFA_example_data$log_x), sfa_fitted[order(SSFA_example_data$log_x)],col="red")

SSFA residuals Moran plot

Description

This function allows to plot the residuals of the object against their spatially lagged values, augmented by reporting the summary of influence measures for the linear relationship between the data and the lag.

Usage

plot_moran(x, main, xlab, ylab, labels, listw, ...)

Arguments

x

an object of class ssfa.
main

an overall title for the plot.
xlab

a label for the x axis.
ylab

a label for the y axis.
labels

character labels for points with high influence measures, if set to FALSE, no labels are plotted for points with large influence.
listw

a listw object from nb2listw (see nb2listw).
...

arguments to be passed to methods, such as graphical parameters (see par).

References

Anselin, L. (1995). Local indicators of spatial association, Geographical Analysis, 27, 93-115.

Anselin, L. (1996). The Moran scatterplot as an ESDA tool to assess local instability in spatial association. pp. 111-125 in M. M. Fischer, H. J. Scholten and D. Unwin (eds) Spatial analytical perspectives on GIS, London, Taylor and Francis.

See Also

moran.plot

Examples

library(ssfa) 
data(SSFA_example_data)
data(Italian_W)

#### SFA and SSFA comparison ###
sfa <- ssfa(log_y ~ log_x, data = SSFA_example_data, data_w=Italian_W, 
             form = "production", par_rho=FALSE)
ssfa <- ssfa(log_y ~ log_x, data = SSFA_example_data, data_w=Italian_W, 
             form = "production", par_rho=TRUE)

moran.test(residuals.ssfa(sfa), sfa$list_w)
moran.test(residuals.ssfa(ssfa), ssfa$list_w)

plot_moran(sfa, listw=sfa$list_w)
plot_moran(ssfa, listw=ssfa$list_w)

SSFA residuals

Description

This function returns the residuals of the fitted SSFA model.

Usage

## S3 method for class 'ssfa'
residuals(object, ...)

Arguments

object

an object of class ssfa.
...

further arguments for methods.

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)
residuals.ssfa(ssfa)

Spatial stochastic frontier estimation

Description

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

log(yi)=log(f(xi;βi))+vi−uilog(y_{i}) = log(f(x_{i};\beta_i)) +v_{i}-u_{i}

ui=ρ∑iwi.ui+ui~u_{i}=\rho \sum_{i}w_{i.}u_{i} + \widetilde{u_{i}}

where yiy_i are the outputs, xix_i the inputs, viv_i the stochastic noise, uiu_{i} the inefficiency term, rhorho the spatial lag, wi.w_{i.} a standardized row of the spatial weights matrix and ui~\widetilde{u_{i}} the stochastic noise of the inefficiency term.

Usage

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".

Value

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 uu 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.

Author(s)

Fusco E. and Vidoli F.

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.

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

Example dataset

Description

The dataset contains the simulated data used by Fusco and Vidoli (2013) to test the model. Data Generating Process (DGP) follows the construction criteria proposed by Banker and Natarajan (2008), also used by Johnson and Kuosmanen (2011), with the addition of a strong spatial correlation in the inefficiency term through a spatial lag parameter and a contiguity matrix (107 Italian provinces contiguity matrix, year 2008).

Usage

data(SSFA_example_data)

Format

A data frame with 107 observations (Italian provinces) and 2 variables:

DMU

the Decision Making Unit name.

log_x

the input vector (already in logarithmic form).

log_y

the output vector (already in logarithmic form).

References

Banker, R., and R. Natarajan (2008). Evaluating Contextual Variables Affecting Productivity using Data Envelopment Analysis. Operations Research 56 (1): 48-58.

Johnson, A., and T. Kuosmanen (2011). One-stage Estimation of the Effects of Operational Conditions and Practices on Productive Performance: Asymptotically Normal and Efficient, Root-n Consistent StoNEZD Method. Journal of Productivity Analysis 36:219-230.

Examples

data(SSFA_example_data)

SSFA summaries

Description

The function print.ssfa is used to display the values of SFA and SSFA estimated coefficients. In particular:

- for SFA the function displays the Intercept, the regressors beta coefficients, the inefficiency variance sigmau2, the stochastic error variance sigmav2 and the total variance sigma2;
- for SSFA the function displays, in addition, the decomposition of the inefficiency variance into sigmau2_dmu and sigmau2_sar, respectively the part of inefficiency variance due to DMU's specificities and to the spatial dependence, and finally, the spatial lag parameter rho.

The function summary.ssfa is used to display the summary results of SFA and SSFA. In particular:

- for SFA the summary shows the estimation of SFA coefficients (Intercept, beta coefficients, sigmau2 and sigmav2) and others useful information as the total variance sigma2, the inefficiency parameter Lambda (sigmau/sigmav), the Moran I statistic, the mean of efficiency, the LR-test and the AIC values;
- for SSFA the summary shows, in addition, the decomposition of the inefficiency variance into sigmau2_dmu and sigmau2_sar and the spatial lag parameter rho.

Usage

## S3 method for class 'ssfa'
print(x, ...)
## S3 method for class 'ssfa'
summary(object, ...)

Arguments

x

an object of class ssfa.
object

an object of class ssfa.
...

further arguments for methods.

Note

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

Anselin, L. (1995). Local indicators of spatial association, Geographical Analysis, 27, 93-115.

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.

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)

print(ssfa)
summary(ssfa)

SSFA inefficiency

Description

This function returns the specific inefficiency of each producer (without local spatial effects) calculated by the Jondrow et al. (JLMS) (1982) formulation modified by using an autoregressive specification in the inefficiency term.

Usage

u.ssfa(object, ...)

Arguments

object

an object of class ssfa.
...

further arguments for methods.

Value

Inefficiency of each producer (without local spatial effects).

References

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.

Jondrow, J., C. A. Knox Lovell, I. S. Materov, and P. Schmidt. (1982). On the Estimation of Technical Inefficiency in the Stochastic Frontier Production Function Model. Journal of Econometrics 19 (2-3): 233-238.

See Also

eff.ssfa

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)
ineff <- u.ssfa(ssfa)