# Statistical Inference Regarding the Radial Measure of Technical Efficiency Routine `teradialbc` performs bias correction of the radial Debrue-Farrell input- or output-based measure of technical efficiency, computes bias and constructs confidence intervals via bootstrapping techniques. ```r teradialbc(formula, data, subset, ref = NULL, data.ref = NULL, subset.ref = NULL, rts = c("C", "NI", "V"), base = c("output", "input"), homogeneous = TRUE, smoothed = TRUE, kappa = NULL, reps = 999, level = 95, core.count = 1, cl.type = c("SOCK", "MPI"), print.level = 1, dots = TRUE) ``` ## 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 `teradial` 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 `teradial` is called. - `subset.ref`: an optional vector specifying a subset of observations to define the technology reference set. - `smoothed`: logical. If TRUE, the reference set is bootstrapped with smoothing; if FALSE, the reference set is bootstrapped with subsampling. - `homogeneous`: logical. Relevant if `smoothed=TRUE`. If TRUE, the reference set is bootstrapped with homogeneous smoothing; if FALSE, the reference set is bootstrapped with heterogeneous smoothing. - `kappa`: relevant if `smoothed=TRUE`. 'kappa' sets the size of the subsample as K^kappa, where K is the number of data points in the original reference set. The default value is 0.7. 'kappa' may be between 0.5 and 1. - `reps`: specifies the number of bootstrap replications to be performed. The default is 999. The minimum is 100. Adequate estimates of confidence intervals using bias-corrected methods typically require 1,000 or more replications. - `level`: sets confidence level for confidence intervals; default is `level = 95`. - `core.count`: positive integer. Number of cluster nodes. If `core.count=1`, the process runs sequentially. See `performParallel` in package `snowFT` for more details. - `cl.type`: Character string that specifies cluster type (see `makeClusterFT` in package `snowFT`). Possible values are 'MPI' and 'SOCK' ('PVM' is currently not available). See `performParallel` in package `snowFT` for more details. - `dots`: logical. Relevant if `print.level>=1`. If TRUE, one dot character is displayed for each successful replication; if FALSE, display of the replication dots is suppressed. - `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 `teradialbc` performs bias correction of the radial Debrue-Farrell input- or output-based measure of technical efficiency, computes bias and constructs confidence intervals via bootstrapping techniques. See Simar and Wilson (1998) tools:::Rd_expr_doi("10.1287/mnsc.44.1.49") , Simar and Wilson (2000) tools:::Rd_expr_doi("10.1080/02664760050081951") , Kneip, Simar, and Wilson (2008) tools:::Rd_expr_doi("10.1017/S0266466608080651") , and references with links below. Models for `teradialbc` 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. If `core.count>=1`, `teradialbc` will perform bootstrap on multiple cores. Parallel computing requires package `snowFT`. By the default cluster type is defined by option `cl.type="SOCK"`. Specifying `cl.type="MPI"` requires package `Rmpi`. On some systems, specifying option `cl.type="SOCK"` results in much quicker execution than specifying option `cl.type="MPI"`. Option `cl.type="SOCK"` might be problematic on Mac system. Parallel computing make a difference for large data sets. Specifying option `dots=TRUE` will indicate at what speed the bootstrap actually proceeds. Specify `reps=100` and compare two runs with option `core.count=1` and `core.count>1` to see if parallel computing speeds up the bootstrap. For small samples, parallel computing may actually slow down the `teradialbc`. Results can be summarized using `summary.npsf`. ## Returns `teradialbc` returns a list of class `npsf` containing the following elements: - **K**: numeric: number of data points. - **M**: numeric: number of outputs. - **N**: numeric: number of inputs. - **rts**: string: RTS assumption. - **base**: string: base for efficiency measurement. - **reps**: numeric: number of bootstrap replications. - **level**: numeric: confidence level for confidence intervals. - **te**: numeric: radial measure (Russell) of technical efficiency. - **tebc**: numeric: bias-corrected radial measures of technical efficiency. - **biasboot**: numeric: bootstrap bias estimate for the original radial measures of technical efficiency. - **varboot**: numeric: bootstrap variance estimate for the radial measures of technical efficiency. - **biassqvar**: numeric: one-third of the ratio of bias squared to variance for radial measures of technical efficiency. - **realreps**: numeric: actual number of replications used for statistical inference. - **telow**: numeric: lower bound estimate for radial measures of technical efficiency. - **teupp**: numeric: upper bound estimate for radial measures of technical efficiency. - **teboot**: numeric: `reps x K` matrix containing bootstrapped measures of technical efficiency from each of `reps` bootstrap replications. - **esample**: logical: returns TRUE if the observation in user supplied data is in the estimation subsample and FALSE otherwise. ## Note Before specifying option `homogeneous` it is advised to preform the test of independence (see `nptestind`). Routine `nptestrts` may help deciding regarding option `rts`. 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") 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") Kneip, A., Simar L., and P.W. Wilson (2008), Asymptotics and Consistent Bootstraps for DEA Estimators in Nonparametric Frontier Models, **Econometric Theory**, 24 , 1663--1697, tools:::Rd_expr_doi("10.1017/S0266466608080651") Simar, L. and P.W. Wilson (1998), Sensitivity Analysis of Efficiency Scores: How to Bootstrap in Nonparametric Frontier Models, **Management Science**, 44 , 49--61, tools:::Rd_expr_doi("10.1287/mnsc.44.1.49") Simar, L. and P.W. Wilson (2000), A General Methodology for Bootstrapping in Nonparametric Frontier Models, **Journal of Applied Statistics**, 27 , 779--802, tools:::Rd_expr_doi("10.1080/02664760050081951") ## Author(s) Oleg Badunenko , Pavlo Mozharovskyi ## See Also `teradial`, `tenonradial`, `tenonradialbc`, `nptestrts`, `nptestind`, `sf` ## Examples ```r ## Not run: require( npsf ) # Prepare data and matrices data( pwt56 ) head( pwt56 ) # Create some missing values pwt56 [49, "K"] <- NA # just to 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 # just to create missing X1 [49, 1] <- NA # just to create missing data( ccr81 ) head( ccr81 ) # Create some missing values ccr81 [64, "x4"] <- NA # just to create missing ccr81 [68, "y2"] <- NA # just to create missing Y2 <- as.matrix( ccr81[ , c("y1", "y2", "y3"), drop = FALSE] ) X2 <- as.matrix( ccr81[ , c("x1", "x2", "x3", "x4", "x5"), drop = FALSE] ) # Compute output-based measures of technical efficiency under # the assumption of CRS (the default) and perform bias-correctiion # using smoothed homogeneous bootstrap (the default) with 999 # replications (the default). t1 <- teradialbc ( y1 + y2 + y3 ~ x1 + x2 + x3 + x4 + x5, data = ccr81) # or just t2 <- teradialbc ( Y2 ~ X2) # Combined formula and matrix t3 <- teradialbc ( Y ~ K + L, data = pwt56, subset = Nu < 10, ref = Y1[-2,] ~ X1[-1,] ) # Compute input-based measures of technical efficiency under # the assumption of VRS and perform bias-correctiion using # subsampling heterogenous bootstrap with 1999 replications. # Choose to report 99 # formed by data points where x5 is not equal 10. # Suppress printing dots. t4 <- teradialbc ( y1 + y2 + y3 ~ x1 + x2 + x3 + x4 + x5, data = ccr81, ref = y1 + y2 + y3 ~ x1 + x2 + x3 + x4 + x5, subset.ref = x5 != 10, data.ref = ccr81, reps = 1999, smoothed = FALSE, kappa = 0.7, dots = FALSE, base = "i", rts = "v", level = 99) # Compute input-based measures of technical efficiency under # the assumption of NRS and perform bias-correctiion using # smoothed heterogenous bootstrap with 499 replications for # all data points. The reference set formed by data points # where x5 is not equal 10. t5 <- teradialbc ( y1 + y2 + y3 ~ x1 + x2 + x3 + x4 + x5, data = ccr81, ref = y1 + y2 + y3 ~ x1 + x2 + x3 + x4 + x5, subset.ref = x5 != 10, data.ref = ccr81, homogeneous = FALSE, reps = 999, smoothed = TRUE, dots = TRUE, base = "i", rts = "n") # =========================== # === Parallel computing === # =========================== # Perform previous bias-correction but use 8 cores and # cluster type SOCK t51 <- teradialbc ( y1 + y2 + y3 ~ x1 + x2 + x3 + x4 + x5, data = ccr81, ref = y1 + y2 + y3 ~ x1 + x2 + x3 + x4 + x5, subset.ref = x5 != 10, data.ref = ccr81, homogeneous = FALSE, reps = 999, smoothed = TRUE, dots = TRUE, base = "i", rts = "n", core.count = 8, cl.type = "SOCK") # Really large data-set data(usmanuf) head(usmanuf) nrow(usmanuf) table(usmanuf$year) # This will take some time depending on computer power data(usmanuf) head(usmanuf) t6 <- teradialbc ( Y ~ K + L + M, data = usmanuf, subset = year >= 1999 & year <= 2000, homogeneous = FALSE, base = "o", reps = 100, core.count = 8, cl.type = "SOCK") ## End(Not run) ```

teradialbc function