Package {spscsfa}

Extract JLMS inefficiency estimates

Description

Extract JLMS inefficiency estimates

Usage

inefficiency(x)

Arguments

Value

A numeric vector of JLMS conditional inefficiency estimates.

Print SPSC-SFA results

Description

Print SPSC-SFA results

Usage

## S3 method for class 'spsc_sfa'
print(x, ...)

Arguments

Value

The original object of class "spsc_sfa", returned invisibly to avoid duplicate printing after printing its summary. The returned object is a list containing the components smooth_coefficients, mle_results, technical_efficiency, inefficiency, model_summary, formula_parsed, and data_info. These components contain the estimated smooth coefficients and recovered intercept, maximum-likelihood results, technical-efficiency estimates and confidence intervals, JLMS inefficiency estimates, model-specification information, parsed formula components, and data-cleaning information.

Philippine Rice Farm Data

Description

Data used to illustrate the application of the semiparametric smooth-coefficient stochastic frontier model.

Usage

data(ricephil)

Format

A data frame with 344 observations and 17 variables.

References

Timothy J. Coelli, D.S. Prasada Rao, Christopher J. O'Donnell, and George E. Battese (2005). An Introduction to Efficiency and Productivity Analysis. Springer, New York.

S. Pandey, P. Masciat, L. Velasco, and R. Villano (1999). Risk analysis of a rainfed rice production system in Tarlac, Central Luzon, Philippines. Experimental Agriculture, 35(2), 225–237.

Extract smooth coefficients

Description

Extract smooth coefficients

Usage

smooth_coef(x)

Arguments

Value

The estimated smooth-coefficient matrix.

Semiparametric Smooth-Coefficient Stochastic Frontier Analysis

Description

Estimates a semiparametric smooth-coefficient stochastic frontier model y_i = \alpha(Z_i) + X_i' \beta(Z_i) + v_i - u(Z_i) following Sun and Kumbhakar (2013). The model allows the coefficients of the parametric part to vary smoothly with a set of nonparametric variables. Technical efficiency and inefficiency are computed using the Battese and Coelli (1988) and Jondrow et al. (1982) methods, respectively. Confidence intervals for technical efficiency are computed using the approach of Horrace and Schmidt (1996).

Usage

spsc_sfa(
  formula,
  data = NULL,
  ezdat = NULL,
  sig_level = 0.05,
  optim_method = "optim_BFGS",
  optim_control = list(maxit = 5000),
  stval = "automatic",
  bandwidth = NULL,
  penalty = 1e+08,
  verbose = TRUE
)

Arguments

Details

The stochastic production frontier model is specified as

y_i = \alpha(Z_i) + X_i' \beta(Z_i) + v_i - u(Z_i),

where y_i can be viewed as the log of output, X_i can be viewed as a vector of the log of inputs, and Z_i is a vector of covariates. \alpha(\cdot) is the intercept, and \beta(\cdot) is the slope vector. Both \alpha(\cdot) and \beta(\cdot) are unknown functions of Z_i. The noise term v_i \sim iid N(0, \sigma_v^2). The inefficiency term is u(Z_i) = \sigma_u(Z_i) \eta_i, where \eta_i \sim iid N^+(0,1). The random variables \eta_i and v_i are assumed to be independent of each other and independent of X and Z.

To guarantee positivity of \sigma_u(Z_i),

\sigma_u(Z_i) = \exp(\delta_0 + \delta_1' Z_i).

These imply that

E(u_i \mid Z_i) = \sigma_u(Z_i) E(\eta_i \mid Z_i) = \sqrt{2/\pi}\,\sigma_u(Z_i) = \sqrt{2/\pi}\,\exp(\delta_0 + \delta_1' Z_i)

and \sigma^2_u(Z_i) = \exp[2(\delta_0 + \delta_1' Z_i)].

The estimation involves two steps. In the first step, estimate \theta(Z_i) (Intercept (theta)), \beta(Z_i), and \varepsilon_i from

y_i = \theta(Z_i) + X_i'\beta(Z_i) + \varepsilon_i,

where \theta(Z_i) = \alpha(Z_i)-E(u_i\mid Z_i) and \varepsilon_i = v_i-[u(Z_i)-E(u_i\mid Z_i)] = E(u_i\mid Z_i) + v_i-u(Z_i).

Then in the second step, use the estimated \varepsilon_i to estimate \delta_0, \delta_1 and \sigma^2_v from

\varepsilon_i = \sqrt{2/\pi}\,\exp(\delta_0+\delta_1'Z_i) + v_i - \exp(\delta_0+\delta_1'Z_i)\eta_i.

Finally, compute E(u_i \mid Z_i) = \sqrt{2/\pi}\,\exp(\delta_0 + \delta_1' Z_i) from estimated \delta_0 and \delta_1. The intercept in the original stochastic production frontier \alpha(Z_i) (Intercept (alpha)) is recovered as

\alpha(Z_i) =\theta(Z_i)+E(u_i\mid Z_i).

See Sun and Kumbhakar (2013, page 306) for more details.

Value

An object of class "spsc_sfa" containing smooth coefficients, MLE results, technical efficiency, JLMS inefficiency, model summary, data information, and the parsed formula.

A list of class "spsc_sfa" containing:

Note

Results can be summarized using summary.spsc_sfa. Extraction functions for smooth coefficients, technical efficiency and inefficiency are available through smooth_coef, te, and inefficiency.

Author(s)

Kai Sun <ksun1@shu.edu.cn>

References

George E. Battese and Tim 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.

William C. Horrace and Peter Schmidt (1996). Confidence statements for efficiency estimates from stochastic frontier models. Journal of Productivity Analysis, 7(2), 257–282.

James Jondrow, C.A. Knox Lovell, Ivan S. Materov, and Peter Schmidt (1982). On the estimation of technical inefficiency in the stochastic frontier production function model. Journal of Econometrics, 19(2–3), 233–238.

Kai Sun and Subal C. Kumbhakar (2013). Semiparametric smooth-coefficient stochastic frontier model. Economics Letters, 120(2), 305–309.

Examples

set.seed(1)
n <- 100
x <- rnorm(n)
z <- runif(n)
ez <- seq(0, 1, length = 1000)
q <- rnorm(n)
y <- cos(z) + sin(z) * x + rnorm(n, 0, 0.1) -
     exp(0.1 + 0.1 * z) * abs(rnorm(n, 0, 1))
dat <- data.frame(output = y, input = x, covariate = z, ineffvar = q)

# covariate enters the nonparametric part and is also treated as inefficiency determinant
result <- spsc_sfa(output ~ input | covariate, data = dat,
                   optim_method = "optim_BFGS", verbose = TRUE)
result$smooth_coefficients
result$mle_results
result$technical_efficiency
result$inefficiency

# This gives the same results: the inefficiency variables default to the nonparametric part
result <- spsc_sfa(output ~ input | covariate | NULL, data = dat,
                   optim_method = "optim_BFGS", verbose = TRUE)

# Use evaluation data (ezdat) for estimating the smooth coefficients
result <- spsc_sfa(output ~ input | covariate, data = dat,
                   ezdat = data.frame(covariate = ez),
                   optim_method = "optim_BFGS", verbose = TRUE)
nrow(result$smooth_coefficients$theta_beta)

# Three-part formula: nonparametric part differs from inefficiency determinants
result <- spsc_sfa(output ~ input | covariate | ineffvar, data = dat,
                   optim_method = "optim_BFGS", verbose = TRUE)

# Customized bandwidth for the nonparametric part
result <- spsc_sfa(output ~ input | covariate | ineffvar, data = dat,
                   optim_method = "optim_BFGS", bandwidth = c(0.1), verbose = TRUE)

# Restrict no inefficiency determinants (intercept only)
result <- spsc_sfa(output ~ input | covariate | 1, data = dat,
                   optim_method = "optim_BFGS", verbose = TRUE)

# More variables
set.seed(1)
n <- 100
x1 <- rnorm(n)
x2 <- rnorm(n)
z1 <- runif(n)
z2 <- runif(n)
y <- cos(z1 + z2) + sin(z1 - z2) * x1 + sin(z1 + z2) * x2 +
     rnorm(n, 0, 0.1) - exp(0.1 + 0.1 * z1 + 0.1 * z2) * abs(rnorm(n, 0, 1))
dat <- data.frame(y, x1, x2, z1, z2)

result <- spsc_sfa(y ~ x1 + x2 | z1 + z2, data = dat)

# z3 is an unordered categorical variable
# spsc_sfa() detects if z3 is a factor
dat$z3 <- factor(rep(sample(c("A", "B", "C", "D", "E")), 20))

# z1, z2, z3 enter the nonparametric part and are also inefficiency determinants
result <- spsc_sfa(y ~ x1 + x2 | z1 + z2 + z3, data = dat)

# z3 is preserved as an unordered categorical variable in the nonparametric part
result$smooth_coefficients$bandwidth

# z3 is converted to dummy variables as inefficiency determinants
result$mle_results$delta_table

# Same as above (inefficiency defaults to nonparametric part)
result <- spsc_sfa(y ~ x1 + x2 | z1 + z2 + z3 | NULL, data = dat)

# Three-part formula: use z3 only as the inefficiency determinant
result <- spsc_sfa(y ~ x1 + x2 | z1 + z2 + z3 | z3, data = dat)

# Use external variables as inefficiency determinants
dat$q1 <- rnorm(n)
dat$q2 <- rnorm(n)
result <- spsc_sfa(y ~ x1 + x2 | z1 + z2 + z3 | q1 + q2, data = dat)

# No inefficiency determinants (intercept only)
result <- spsc_sfa(y ~ x1 + x2 | z1 + z2 + z3 | 1, data = dat, verbose = TRUE)

# Application with real data (ricephil)
## Not run: 

data(ricephil)

ricephil$FARMERCODE <- as.factor(ricephil$FARMERCODE)

# this example shows categorical variable handling
result <- spsc_sfa(PROD ~ AREA + LABOR + NPK | EDYRS + FARMERCODE | FARMERCODE,
                   data = ricephil, optim_method = "optim_BFGS", verbose = TRUE)

# FARMERCODE is preserved as an unordered categorical variable in the nonparametric part
result$smooth_coefficients$bandwidth

# FARMERCODE is converted to dummy variables as inefficiency determinants
result$mle_results$delta_table

## End(Not run)

Summary method for SPSC-SFA

Description

Summary method for SPSC-SFA

Usage

## S3 method for class 'spsc_sfa'
summary(object, ...)

Arguments

Value

See print.spsc_sfa.

Extract technical-efficiency scores

Description

Extract technical-efficiency scores

Usage

te(x)

Arguments

Value

A numeric vector of technical-efficiency estimates.