# Nonradial Measure of Technical Efficiency, the Russell Measure 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. ```r 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`. ## Returns `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`. ## References Badunenko, O. and Mozharovskyi, P. (2016), Nonparametric Frontier Analysis using Stata, **Stata Journal**, 16 3, 550--89, tools:::Rd_expr_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**, 71 3, 517--527, tools:::Rd_expr_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, tools:::Rd_expr_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, tools:::Rd_expr_doi("10.1017/CBO9780511551710") ## Author(s) Oleg Badunenko , Pavlo Mozharovskyi ## See Also `teradial`, `teradialbc`, `tenonradialbc`, `nptestrts`, `nptestind`, `sf`, `summary.npsf` for printing summary results ## Examples ```r 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) ```

tenonradial function