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 <fusco_elisa@libero.it>

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.

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

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

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

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 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 u.

Usage

eff.ssfa(object, ...)

Arguments

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

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)

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

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

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

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(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.

Usage

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

Arguments

Value

ssfa returns the following objects of class ssfa:

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.

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

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

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

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)