{smcl}
{* *! version 1.0.1 5jul2011}{...}
{cmd:help orderm}
{hline}
{title:Title}
{p2colset 5 20 22 2}{...}
{phang}
{bf:orderm} {hline 2} Order-m efficiency analysis{p_end}
{p2colreset}{...}
{title:Syntax}
{p 8 17 2}
{cmd:orderm}
{it:{help varlist:varlist1}} = {it:{help varlist:varlist2}} {ifin}, [{cmd:}{it:{help orderalpha##options:options}}]
{synoptset 28 tabbed}{...}
{marker technology_definition}{...}
{synopthdr :technology_definition}
{synoptline}
{syntab :Model}
{synopt :{it:{help varlist:varlist1}}}list of {hi:inputs}{p_end}
{synopt :{it:{help varlist:varlist2}}}list of {hi:outputs}{p_end}
{synoptset 28 tabbed}{...}
{synopthdr :options}
{synoptline}
{syntab :Main}
{synopt :{opth {ul on}dmu{ul off}(varname)}}identifier; default is observation number {it:_n}{p_end}
{space 6}{cmd:{ul on}ort{ul off}(}{ul on}{it:i}{ul off}{it:nput}|{ul on}{it:o}{ul off}{it:utput}{cmd:)} {space 9} consider{it:input} or {it:output} oriented efficiency; default is {opt ort(input)}
{synopt :{opt {ul on}m{ul off}(#)}}set size of reference sample; default is {opt m}(ceil({it:N}^(2/3)){p_end}
{synopt :{opt d:raws(#)}}set number of re-sampling replications; default is {opt draws(200)}{p_end}
{syntab :SE/Bootstrap}
{synopt :{opt boot:strap}}perform bootstrap using 50 replications{p_end}
{synopt :{opt reps:(#)}}set (temporary) number {it:#} of bootstrap replications and perform bootstrap{p_end}
{synopt :{opt tun:e(#)}}set tuning parameter for sub-sampling bootstrap; values within the [0.5,1] interval are allowed{p_end}
{syntab :Reporting}
{synopt :{opt lev:el(#)}}set confidence level; default is {opt level(95)}{p_end}
{space 6}{cmd:{ul on}tab{ul off}le(}{ul on}{it:f}{ul off}{it:ull}|{ul on}{it:s}{ul off}{it:cores}{cmd:)} {space 8} display table of results
{synopt :{opt dot:s(1|2)}}display replication/loop dots{p_end}
{synopt :{opt inv:ert}}report reciprocal of output-oriented efficiency scores{p_end}
{syntab :Generate}
{synopt :{opt gen:erate(newvarlist)}}supply names for new variables, containing efficiency scores, ranks, and pseudo-reference dmus{p_end}
{synopt :{opt repl:ace}}replace existing variables in {it:newvarlist}{p_end}
{synopt :{opt nog:enerate}}do not create new variables containing results{p_end}
{synoptline}
{p2colreset}{...}
{p 4 6 2}
{opt weights} are not allowed; see {help weight}.{p_end}
{p 4 6 2}{cmd:bootstrap}, {cmd:by}, and {cmd:svy} are not allowed; see {help prefix}.{p_end}
{title:Description}
{pstd}
{cmd:orderm} computes non-parametric order-m efficiency scores for decision making units (dmus) as proposed by Daraio and Simar (2007: 72).
Order-m efficiency (Cazals et al., 2002) generalizes the free disposal hull (FDH) approach to efficiency measurement (Deprins et al., 1984). For {opt m(#)}
approaching infinity {cmd:orderm} coincides with FDH. Unlike FDH, which envelopes all data points by a non-convex production possibility frontier, {cmd:orderm} is a partial frontier approach that
allows for super-efficient units that are located beyond the estimated frontier. I.e. rather than performing FDH efficiency analysis using the entire sample as reference,
{cmd:orderm} uses artificial reference samples of size {opt m(#)}, which are randomly drawn with replacement from the peer dmus in the original data. Drawing the artificial sample
is repeated {opt draws(#)} times and order-m efficiency scores are estimated as averages of FDH-like efficiency scores. Depending on the composition of the artificial reference sample,
a dmu may or may not serve as its own reference. This {hline 1} unlike FHD where a dmu is always available as its own peer {hline 1} allows for super-efficient
dmus. I.e. (input-oriented) efficiency scores may exceed the value of one. The number of dmus is limited to the value of {help matsize:matsize}; for large values for {opt m(#)} and/or if bootstrapping is
specified, the maximum allowed sample size may be smaller than this.
{title:Technology Definition}
{dlgtab:Model}
{phang}
{opt varlist1} specifies inputs to the analyzed production process. At least one input-variables is required. Any variable in {it:varlist1} needs to be numeric and strictly positive. Dmus with missing or non-positive values in
{it:varlist1} are dropped.
{phang}
{opt varlist2} specifies outputs from the analyzed production process. At least one output-variables is required. Any variable in {it:varlist2} needs to be numeric and strictly positive. Dmus with missing or non-positive values in
{it:varlist2} are dropped. {it:varlist2} may not share any variable with {it:varlist1}.
{marker options}{...}
{title:Options}
{dlgtab:Main}
{phang}
{opt dmu(varname)} specifies an identifier for the considered dmus. {it:varname} must uniquely identify dums. It may be either a numeric or a string variable.
If no identifer is specified, the observation number {it:_n} is used. In oder to make estimation results easily accessible and result tables informative,
it is recommended to choose an informative variable such as the dmus' real names.
{phang}
{opt ort(input|output)} specifies whether {it:input} or {it:output} oriented efficiency is computed. For the former, inefficiency is
defined in terms of possible proportional reduction in input consumption. For the latter, inefficiency is defined in terms of possible
proportional increase in output generation. For {opt ort(input)} efficiency scores are smaller than one for inefficient dmus,
for {opt ort(output)} efficiency scores are greater than one for inefficient dmus, unless the {opt invert} option is specified. Efficient dmus
in either case are indicated by efficiency scores taking the value one. Super-efficient dmus located beyond the estimated production
possibility frontier, exhibit input-oriented efficiency greater than one, and output-oriented efficiency smaller than unity.
{phang}
{opt m(#)} specifies the size of the artificial reference sample. The default is {opt m}(ceil({it:N}^(2/3))). Non-integer and non-positive values are not allowed. Most
applications choose values substantially smaller than {it:N}. Note: even for {opt m(N)}, {cmd:orderm} does not yield results for FDH efficiency analysis. This rather requires
{opt m(#)} to approach infinity. Yet {hline 1} rather than by choosing a very large value for {opt m(#)} {hline 1} FDH efficiency analysis can by carried out
more efficiently using {help orderalpha:orderalpha}.
{phang}
{opt draws(#)} specifies the number of re-sampling repetitions. The default is {opt draws(200)}, as suggested by Daraio and Simar (2007). Yet, depending
on the data at hand, making estimated efficiency scores converge may require values that substantially exceed the default. Non-integer and non-positive
values are not allowed.
{dlgtab:SE/Bootstrap}
{phang}
{opt bootstrap} invokes bootstrapping using 50 boostrap replications. Unless SEs are definitely required, users are strongly advise not to
request bootstrapping for large (and even moderately sized) samples. Due to nested re-sampling, computing time required by bootstrapping may become excessive. One may
also consider {help orderalpha:orderalpha} as alternative. If neither {opt bootstrap} nor {opt reps(#)} is specified, {cmd:orderm} does not compute standard
errors for the estimated efficiency scores. The bootstrap will fail in determining non-zero SEs for dmus, for which no peers are available in the sample,
apart from the dmu itself.
{phang}
{opt reps(#)} is equivalent to {opt bootstrap}, besides allowing for choosing the number of bootstrap repetitions {it:#}.
{phang}
{opt tune(#)} determines the size of the bootstrap samples as int({it:N}^#). Values within the [0.5,1] interval are allowed. Subsampling is applied in order to
account for the naive bootstrap being inconsistent in a boundary estimation framework. The boundary nature of the estimation problem
vanishes as {opt m(#)} departs from infinity. For small values of {opt m(#)} one may apply the naive bootstrap,
i.e. {opt tune(1)}. The default is {opt tune}((2+exp(-m/N))/3), which is equal to 2/3 for FDH.
{dlgtab:Reporting}
{phang}
{opt level(#)}; see {helpb estimation options##level():[R] estimation options}.
{phang}
{opt table(full|scores)} invokes displaying a results table. For {opt table(scores)} estimated efficiency scores
are displayed as if they were regression coefficients. For {opt table(full)} efficiency ranks and pseudo-reference dmus are also displayed. {cmd:orderm}
may generate a huge table as {it:N} scores are computed. Hence, suppressing table display
is the default. {opt table(full)} is not allowed for {it:N} > 2994 and cannot be re-displayed by typing {cmd:orderm} without arguments.
{phang}
{opt dots(1|2)} invokes a display of replication dots and loop dots. For {opt dots(1)} one dot
character is displayed for each bootstrap replication. For {opt dots(2)} one dot character is also displayed for
each dmu being analyzed. Type {opt 2} dots are not displayed during bootstrap replications.
{phang}
{opt invert} makes output-oriented efficiency being reported analogously to input-oriented efficiency by taking the reciprocal. That
is, with {opt invert} specified, inefficient dmus exhibit efficiency scores smaller than one, irrespective of how {opt ort()} is specified. {opt invert}
has no effect on input-oriented efficiency.
{dlgtab:Generate}
{phang}
{opt generate(newvarlist)} specifies the names of a new variables containing estimation results. {it:newvarlist} may
consist of up to three names. {it:newvar1} denotes estimated efficiency scores, {it:newvar2} denotes efficiency
ranks, and {it:newvar3} denotes the name of the pseudo-reference dmu. If {hline 1} because of ties in the data {hline 1} for some dums more than one
pseudo-reference dmu is identified, further variables {it:newvar3}_2, {it:newvar3}_3, ... are created. If {opt generate(newvarlist)}
is not specified or less than three names are assigned, default names are {it:_om_ort_m}, {it:_omrank_ort_m}, and
{it:_omref_ort_m}.
{phang}
{opt replace} specifies that existing variables named {it:newvar1}, {it:newvar2}, or {it:newvar3} may be replaced.
{phang}
{opt nogenerate} specifies that results are not saved to new variables.
{title:Examples}
{pstd}Order-m(25) output-oriented efficiency{p_end}
{phang2}{cmd:. orderm capital labor energy = durables perishables, dmu(firm) ort(output) m(25) gen(effi rank ref)}{p_end}
{pstd}Order-m(10) input-oriented efficiency with boostrap{p_end}
{phang2}{cmd:. orderm capital labor energy = durables perishables, dmu(firm) m(10) reps(50) dots(2) gen(effi rank ref) replace}{p_end}
{pstd}After orderm with bootstrapping, stata's testing routines can be used as usual{p_end}
{phang2}{cmd:. test _b[firm:Boogle]-_b[firm:Macrosoft]=0}{p_end}
{title:Saved results}
{pstd}
{cmd:orderalpha} saves the following in {cmd:e()}:
{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: Scalars}{p_end}
{synopt:{cmd:e(N)}}number of dmus{p_end}
{synopt:{cmd:e(m)}}value of {opt m(#)}{p_end}
{synopt:{cmd:e(draws)}}value of {opt draws(#)}{p_end}
{synopt:{cmd:e(inputs)}}number of inputs{p_end}
{synopt:{cmd:e(outputs)}}number of outputs{p_end}
{synopt:{cmd:e(efficient)}}share of efficient dmus{p_end}
{synopt:{cmd:e(super)}}share of super-efficient dmus{p_end}
{synopt:{cmd:e(mean_e)}}mean estimated efficiency{p_end}
{synopt:{cmd:e(med_e)}}median estimated efficiency{p_end}
{synopt:{cmd:e(level)}}confidence level{p_end}
{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: Macros}{p_end}
{synopt:{cmd:e(cmd)}}{cmd:orderm}{p_end}
{synopt:{cmd:e(cmdline)}}command as typed{p_end}
{synopt:{cmd:e(title)}}{cmd:Order-m efficiency analysis}{p_end}
{synopt:{cmd:e(dmuid)}}{it:varname} (name of dmu-identifier){p_end}
{synopt:{cmd:e(model)}}{cmd:Order-m}{p_end}
{synopt:{cmd:e(saved)}}names of variables saved (not saved for option {opt nogenerate}){p_end}
{synopt:{cmd:e(table)}}{cmd:scores}, {cmd:full}, or {cmd:no}{p_end}
{synopt:{cmd:e(invert)}}either {opt inverted} or {opt notinverted} (not saved for {opt ort(input)}){p_end}
{synopt:{cmd:e(ort)}}either {opt input} or {opt output}{p_end}
{synopt:{cmd:e(outputlist)}}{it:varlist2} (list of outputs){p_end}
{synopt:{cmd:e(inputlist)}}{it:varlist1} (list of inputs){p_end}
{synopt:{cmd:e(properties)}}either {opt b} or {opt b V}{p_end}
{synopt:{cmd:e(depvar)}}{cmd:dmu}{p_end}
{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: Matrices}{p_end}
{synopt:{cmd:e(b)}}vector of estimated efficiency scores ({it:colnames} are of the form {it: varname:value_of_varname}){p_end}
{synopt:{cmd:e(ranks)}}vector of efficiency ranks ({it:colnames} are of the form {it: varname:value_of_varname}){p_end}
{synopt:{cmd:e(reference)}}matrix of names of pseudo-reference dums (not saved if {it:varname} is a string variable){p_end}
{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: Functions}{p_end}
{synopt:{cmd:e(sample)}}marks estimation sample{p_end}
{p2colreset}{...}
{pstd}
{cmd:orderm, boot reps(#)} additionally saves the following in {cmd:e()}:
{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: Scalars}{p_end}
{synopt:{cmd:e(N_reps)}}number of bootstrap replications{p_end}
{synopt:{cmd:e(tune)}}value of tuning parameter{p_end}
{synopt:{cmd:e(N_bs)}}size of bootstrap samples{p_end}
{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: Macros}{p_end}
{synopt:{cmd:e(vce)}}{cmd:bootstrap}{p_end}
{synopt:{cmd:e(vcetype)}}{cmd:Bootstrap}{p_end}
{synoptset 20 tabbed}{...}
{p2col 5 20 24 2: Matrices}{p_end}
{synopt:{cmd:e(V)}}bootstrap variance-covariance matrix for estimated efficiency scores{p_end}
{synopt:{cmd:e(bias)}}estimated biases{p_end}
{synopt:{cmd:e(reps)}}number of nonmissing results{p_end}
{synopt:{cmd:e(b_bs)}}bootstrap estimates{p_end}
{title:References}
{pstd}
Cazals, C., J.P. Florens and L. Simar (2002). Nonparametric frontier estimation: a robust approach. {it:Journal of Econometrics} 106, 1–25.
{pstd}
Daraio, C. and L. Simar (2007). {it:Advanced robust and nonparametric methods in efficiency analysis: Methodology and applications}. Springer, New York.
{pstd}
Deprins, D., L. Simar and H. Tulkens (1984). Measuring laborefficiency in post offices, in: Marchand, M., P. Pestieau, H. Tulkens, (eds.),
{it:The Performance of Public Enterprises: Concepts and Measurement}. Elsevier, Amsterdam, 243-267.
{title:Also see}
{psee}
Manual: {manlink R frontier}
{psee}
{space 2}Help: {manhelp frontier R:frontier}{break}
{psee}
Online: {helpb dea}, {helpb orderalpha}{p_end}
{title:Author}
{psee}
Harald Tauchmann{p_end}
{psee}
Rheinisch-Westfälisches Institut für Wirtschaftsforschung (RWI){p_end}
{psee}
Essen, Germany{p_end}
{psee}
E-mail: harald.tauchmann@rwi-essen.de
{p_end}
{title:Disclaimer}
{pstd} This software is provided "as is" without warranty of any kind, either expressed or implied. The entire risk as to the quality and
performance of the program is with you. Should the program prove defective, you assume the cost of all necessary servicing, repair or
correction. In no event will the copyright holders or their employers, or any other party who may modify and/or redistribute this software,
be liable to you for damages, including any general, special, incidental or consequential damages arising out of the use or inability to
use the program.
{p_end}
{title:Acknowledgements}
{pstd}
This work has been supported in part by the Collaborative Research Center "Statistical Modelling of
Nonlinear Dynamic Processes" (SFB 823) of the German Research Foundation (DFG).
{p_end}