tenonradial {npsf}

Nonradial Measure of Technical Efficiency, the Russell Measure

Description

Routine tenonradial uses reduced linear programming to compute the nonradial output- or input-based measure of technical efficiency, which is known as the Russell measure. In input-based nonradial efficiency measurement, this measure allows for non-proportional/different reductions in each positive input, and this is what permits it to shrink an input vector all the way back to the efficient subset. In output-based nonradial efficiency measurement, the Russell measure allows for non-proportional/different expansions of each positive output.

Usage

tenonradial(formula, data, subset,
 rts = c("C", "NI", "V"), 
 base = c("output", "input"), 
 ref = NULL, data.ref = NULL, subset.ref = NULL,
 full.solution = TRUE,
 print.level = 1)

Arguments

formula

an object of class “formula” (or one that can be coerced to that class): a symbolic description of the model. The details of model specification are given under ‘Details’.

data

an optional data frame containing the variables in the model. If not found in data, the variables are taken from environment (formula), typically the environment from which tenonradial is called.

subset

an optional vector specifying a subset of observations for which technical efficiency is to be computed.

rts

character or numeric. string: first letter of the word “c” for constant, “n” for non-increasing, or “v” for variable returns to scale assumption. numeric: 3 for constant, 2 for non-increasing, or 1 for variable returns to scale assumption.

base

character or numeric. string: first letter of the word “o” for computing output-based or “i” for computing input-based technical efficiency measure. string: 2 for computing output-based or 1 for computing input-based technical efficiency measure

ref

an object of class “formula” (or one that can be coerced to that class): a symbolic description of inputs and outputs that are used to define the technology reference set. The details of technology reference set specification are given under ‘Details’. If reference is not provided, the technical efficiency measures for data points are computed relative to technology based on data points themselves.

data.ref

an optional data frame containing the variables in the technology reference set. If not found in data.ref, the variables are taken from environment(ref), typically the environment from which tenonradial is called.

subset.ref

an optional vector specifying a subset of observations to define the technology reference set.

full.solution

logical. The detailed solution is returned. See value section.

print.level

numeric. 0 - nothing is printed; 1 - print summary of the model and data. 2 - print summary of technical efficiency measures. 3 - print estimation results observation by observation. Default is 1.

Details

Routine tenonradial computes the nonradial output- or input-based measure of technical efficiency under assumption of constant, non-increasing, or variable returns to scale technology. The details of the estimator can be found e.g., in Färe, Grosskopf, and Lovell (1994) or Badunenko and Mozharovskyi (2020).

Models for tenonradial are specified symbolically. A typical model has the form outputs ~ inputs, where outputs (inputs) is a series of (numeric) terms which specifies outputs (inputs). The same goes for reference set. Refer to the examples.

Results can be summarized using summary.npsf.

Value

tenonradial returns a list of class npsf containing the following elements:

model

string: model name.

K

numeric: number of data points for which efficiency is estimated.

M

numeric: number of outputs.

N

numeric: number of inputs.

Kref

numeric: number of data points in the reference.

rts

string: RTS assumption.

base

string: base for efficiency measurement.

te

numeric: nonradial measure (Russell) of technical efficiency.

te.detail

numeric: K x ncol matrix containing thetas or lambdas for ncol outputs (output-based) or inputs (input-based). ncol is M for output- and N for input-based efficiency measurement.

intensity

numeric: K x Kref matrix containing the intensity variables z. These can be used to identify peers.

esample

logical: returns TRUE if the observation in user supplied data is in the estimation subsample and FALSE otherwise.

esample.ref

logical: returns TRUE if the observation in the user supplied reference is in the reference subsample and FALSE otherwise.

Note

In case of one input (output), the input (output)-based Russell measure is equal to Debrue-Farrell (teradial) measure of technical efficiency.

Results can be summarized using summary.npsf.

Author(s)

Oleg Badunenko <[e-mail]>, Pavlo Mozharovskyi <[e-mail]>

References

Badunenko, O. and Mozharovskyi, P. (2016), Nonparametric Frontier Analysis using Stata, Stata Journal, 163, 550–89, doi: 10.1177/1536867X1601600302

Badunenko, O. and Mozharovskyi, P. (2020), Statistical inference for the Russell measure of technical efficiency, Journal of the Operational Research Society, 713, 517–527, doi: 10.1080/01605682.2019.1599778

Färe, R. and Lovell, C. A. K. (1978), Measuring the technical efficiency of production, Journal of Economic Theory, 19, 150–162, doi: 10.1016/0022-0531(78)90060-1

Färe, R., Grosskopf, S. and Lovell, C. A. K. (1994), Production Frontiers, Cambridge U.K.: Cambridge University Press, doi: 10.1017/CBO9780511551710

See Also

teradial, teradialbc, tenonradialbc, nptestrts, nptestind, sf, summary.npsf for printing summary results

Examples

require( npsf )

# Prepare data and matrices

data( pwt56 )
head( pwt56 )

# Create some missing values

pwt56 [49, "K"] <- NA # create missing

Y1 <- as.matrix ( pwt56[ pwt56$year == 1965, c("Y"), drop = FALSE] )
X1 <- as.matrix ( pwt56[ pwt56$year == 1965, c("K", "L"), drop = FALSE] )

X1 [51, 2] <- NA # create missing
X1 [49, 1] <- NA # create missing

data( ccr81 )
head( ccr81 )

# Create some missing values

ccr81 [64, "x4"] <- NA # create missing
ccr81 [68, "y2"] <- NA # create missing

Y2 <- as.matrix( ccr81[ , c("y1", "y2", "y3"), drop = FALSE] )
X2 <- as.matrix( ccr81[ , c("x1", "x2", "x3", "x4", "x5"), drop = FALSE] )

# Computing without reference set

# Using formula

# Country is a categorical variable, so nonradial gives error message

# t1 <- tenonradial ( Country ~ K + L, data = pwt56 )

# for computing the efficiencies of countries in 1965 
# with technology reference set is defined by observations in 1965
# (that same sample of countries)

t2 <- tenonradial ( Y ~ K + L, data = pwt56, rts = "v", 
base = "in", print.level = 2)

# Using a subset

t3 <- tenonradial ( Y ~ K + L, data = pwt56, subset = year == 1965,
	rts = "VRS", base = "in", print.level = 3 )

t4 <- tenonradial ( Y ~ K + L, data = pwt56, subset = Nu < 10,
	rts = "vrs", base = "I" )

t5 <- tenonradial ( Y ~ L, data = pwt56, subset = Nu < 10, rts = "v" )

# Multiple outputs

t8 <- tenonradial ( y1 + y2 + y3 ~ x1 + x2 + x3 + x4 + x5, data = ccr81,
	rts = "v", base = "i" )

# Using a subset

t7 <- tenonradial ( y1 + y2 + y3 ~ x1 + x2 + x3 + x4 + x5, data = ccr81,
	subset = x5 != 22, rts = "n", base = "o" )

# Computation using matrices

t9 <- tenonradial ( Y1 ~ X1, rts = "v", base = "i" )

# Define subsets on a fly

t10 <- tenonradial ( Y1[-1,] ~ X1[-2,1] )
t11 <- tenonradial ( Y1[-3,] ~ X1[-1,], rts = "v", base = "o" )

# Multiple outputs

t12 <- tenonradial ( Y2 ~ X2 )
t13 <- tenonradial ( Y2[-66,] ~ X2[-1, -c(1,3)] )


# Computing with reference set

# Using formula

# For computing the efficiencies of countries with order number
# less than 10 with technology reference set defined by countries
# with order number larger than 10 and smaller than 11 (in effect 
# no reference set, hence warning) type

t14 <- tenonradial ( Y ~ K + L, data = pwt56, subset = Nu < 10, 
	ref = Y ~ K + L, data.ref = pwt56,
	subset.ref = Nu > 10 & Nu < 11 ) # warning

# For computing the efficiencies of countries with order number
# less than 10 with technology reference set defined by countries 
# with order number larger than 10 and smaller than 15 type

t15 <- tenonradial ( Y ~ K + L, data = pwt56, subset = Nu < 10, ref = Y ~ K + L, 
	data.ref = pwt56, subset.ref = Nu > 10 & Nu < 15 )

# For computing the efficiencies of countries in 1965 
# with technology reference set is defined by observations in both
# 1965 and 1990 (all) type
	
t16 <- tenonradial ( Y ~ K + L, data = pwt56, subset = year == 1965,
	rts = "v", base = "i", 
	ref = Y ~ K + L, data.ref = pwt56 )

# For computing the efficiencies of countries in 1990
# with technology reference set is defined by observations in 1965
# type

t17 <- tenonradial ( Y ~ K + L, data = pwt56, subset = year == 1990, 
	ref = Y ~ K + L, data.ref = pwt56, subset.ref = year == 1965 )

# Using matrices

t18 <- tenonradial ( Y1[-1,] ~ X1[-2,], ref = Y1[-2,] ~ X1[-1,] )

# error: not equal number of observations in outputs and inputs

# t19 <- tenonradial ( Y1[-1,] ~ X1[-(1:2),], 
# ref = Y1[-2,] ~ X1[-1,1] )

# Combined formula and matrix

# error: not equal number of inputs in data and reference set

# t20 <- tenonradial ( Y ~ K + L, data = pwt56, subset = Nu < 10,
# ref = Y1[-2,] ~ X1[-1,1] )

t21 <- tenonradial ( Y ~ K + L, data = pwt56, subset = Nu < 10, 
	ref = Y1[-2,] ~ X1[-1,] )

## Not run: 

# Really large data-set

data(usmanuf)
head(usmanuf)

nrow(usmanuf)
table(usmanuf$year)

# This will take some time depending on computer power

t22 <- tenonradial ( Y ~ K + L + M, data = usmanuf, 
	subset = year >= 1995 & year <= 2000 ) 

# Summary

summary ( t22$te )

# Write efficiencies to the data frame:

usmanuf$te_nonrad_crs_out[ t22$esample ] <- t22$te

head(usmanuf, 17)


## End(Not run)