orderalpha: non-parametric order- efficiency analysis for stata · 2011. 7. 19. · measuring...
TRANSCRIPT
orderalpha: non-parametric order-α
Efficiency Analysis for Stata
Harald Tauchmann
Rheinisch-Westfälisches Institut für Wirtschaftsforschung (RWI)
July 1st, 2011
2011 German Stata Users Group Meeting, University of Bamberg
Introduction Analyzing Efficiency
Introduction
I Countless empirical efficiency analyses in economicsI Decision Making Units/production units (DMUs): firms,
states, universities, etc.
I Measuring distance to Production Possibility FrontierI alternatively: cost- or profit-frontier
I Output-orientated and Input-orientated efficiency
1. How much additional output (y1, . . . , yL) can be producedleaving inputs (x1, . . . , xM) unchanged?
2. How much input (x1, . . . , xM) can be saved leaving outputs(y1, . . . , yL) unchanged?
I Prime purpose: estimating efficiency scores θi forindividual DMUs
I (Input-oriented) efficiency score (0,1]: factor by whichinput consumption can proportionally be reduced
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Measuring Efficiency Parametric vs. Non-Parametric Approaches
Parametric Approaches: Stochastic Frontier Models
I Introduced by Aigner et al. (1977)
→ stata command frontier
I Production- (cost-, profit-) function estimated
I Linear regression model augmented by additionalnon-positive (reps. non-negative) error term ηi
I Maximum-likelihood estimation
I Distributional assumptions for ηi and υi required
log(yi) = β0 +M∑
m=1
βMxMi + υi − ηi with ηi ≥ 0
I (Output-oriented) technical efficiency score computed as:
θ̂SFi = E(exp (−ηi) |υi − ηi)
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Measuring Efficiency Parametric vs. Non-Parametric Approaches
Non-Parametric Approaches
1. Data envelopment Analysis (DEA)
I Introduced by Charnes et al. (1978)
→ stata ado-file dea (Ji & Lee, 2010)I Linear programming approachI Envelopes data by piecewise-linear convex hullI Solution for θ (input-oriented) efficiency score θ̂DEA
i :
minθ,λ
θ subject to
θxmi −N∑
j=1
λjxmj ≥ 0 m = 1, ...,M
N∑j=1
λjylj − yli ≥ 0 l = 1, ..., L
λj ≥ 0 ∀ j
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Measuring Efficiency Parametric vs. Non-Parametric Approaches
Non-Parametric Approaches II
2. Free Disposal Hull (FDH)
I Introduced by Deprins et al. (1984)
I Based on principle of weak dominanceI DMU i compared to those DMUs that produce at least the
same amount of any outputI DMU with minimal input use serves as reference
I Envelopes data by piecewise linear non-convex (step) hull
I (Input-oriented) efficiency scores computed as:
θ̂FDHi = min
j=1,...,N|ylj≥ yli ∀ l
{max
m=1,...,M
{xmj
xmi
}}
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Measuring Efficiency Parametric vs. Non-Parametric Approaches
Parametric vs. Non-Parametric Approaches
1. Parametric Approach (shortcomings):I Relies on distributional assumptionsI Functional form for production technology requiredI Production function ill-suited regression model
(endogeneity of inputs)I Accommodates only single-output technologies
2. Non-Parametric Approach (shortcomings):I No well-defined data generating process
(→ recent advances)I Deterministic approach⇒ Extremely vulnerable to outliers and measurement error
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Partial Frontier Approaches Generalizing FDH
Partial Frontier Approaches
I Sensitivity to outliers reduced by allowing forsuper-efficient DMUs
I Super-efficient DMUs located beyond production-possibilityfrontier
I partial frontier envelopes just a sub-sample of the data
I Super-efficiency: (input-oriented) efficiency score > 1
Partial frontier approaches that generalize FDH:
1. Order-m efficiency (Cazals et al., 2002)
2. Order-α efficiency (Aragon et al., 2005)
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Partial Frontier Approaches Order-m Efficiency
Order-m Efficiency
I Adds a ‘layer of randomness’ to FDH
I Series of FDH analyses using an randomly drawnsup-samples of size m
I DMU may or may not serve as its own peer
I Re-sampling repeated D times
I θ̂OMi : average of D efficiency sores
I Shortcoming: very time consuming (for large data sets)
→ Rules virtually out statistical inference based onbootstrapping
→ Determining appropriate value for m may require tryingnumerous values
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Partial Frontier Approaches Order-α Efficiency
Order-α EfficiencyI Chooses [100− α]th (αth) percentile, with 0 ≤ α ≤ 100,
rather an minimum (maximum) as efficiency benchmarkI Not min. input consumption, but [100− α]th percentile
(among peer-DMUs) serves as referenceI FDH represent special case of order-α (for α = 100)I No re-sampling→ less time consuming than order-m
→ New stata ado-file orderalpha
Order-α Input-Oriented Efficiency:
θ̂OAinput
i = P100−αj=1,...,N|ylj≥ yli ∀ l
{max
m=1,...,M
{xmj
xmi
}}Order-α Output-Oriented Efficiency:
θ̂OAoutput
i = Pαj=1,...,N|xmj≤ xmi ∀m
{min
l=1,...,L
{ylj
yli
}}Tauchmann (RWI) orderalpha July 1st, 2011 9 / 20
Partial Frontier Approaches Graphical Illustration
Illustration for Single-Input Single-Output case
(districts-level health production in Germany)
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Partial Frontier Approaches Graphical Illustration
Illustration for Single-Input Single-Output case
(districts-level health production in Germany)
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Partial Frontier Approaches Graphical Illustration
Illustration for Single-Input Single-Output case
(districts-level health production in Germany)
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Partial Frontier Approaches Graphical Illustration
Illustration for Single-Input Single-Output case
(districts-level health production in Germany)
Tauchmann (RWI) orderalpha July 1st, 2011 10 / 20
Partial Frontier Approaches Graphical Illustration
Illustration for Single-Input Single-Output case
(districts-level health production in Germany)
Tauchmann (RWI) orderalpha July 1st, 2011 10 / 20
Partial Frontier Approaches Graphical Illustration
Illustration for Single-Input Single-Output case
(districts-level health production in Germany)
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Order-α for stata Syntax
orderalpha: Syntax
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Order-α for stata Application
Application: Regional Health Production in Bavaria
I Decision making units:I Districts (‘kreis’) in BavariaI Cross section: year 2004I # of observations: 96
I Inputs:
i. Resident medical specialists per 100 000 inhabitants(‘specialists’)
ii. General practitioners per 100 000 inhabitants (‘gps’)iii. Hospital beds per 10 000 inhabitants (‘beds’)
I Outputs:
i. Inverse normalized (to district demographics) mortality(‘survival’)
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Order-α for stata Application
orderalpha: Screen Shot (regional health production in Bavaria)
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Order-α for stata Saved Results
orderalpha: Saved Results
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Order-α based outlier-detection Daraio and Simar
Order-α based outlier-detection
Idee proposed by Daraio & Simar (2007):
I Increasing the value of α reduces number of DMUsclassified as “super-efficient”
I In the absence of outliers: share of super-efficient DMUsshould decrease smoothly
I Discontinuity points at presence of outliers
⇒ DMUs still classified “super-efficient” for α ≥ αdisc (point ofdiscontinuity) most likely outliers
⇒ To be excluded from efficiency analysis applying FDH orDEA
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Order-α based outlier-detection oaoutlier
Implementing approach of Daraio & Simar (2007)
oaoutlier:
I Carries out series of order-α analyses
I Plots share of super-efficient units against α
I Suggests two local and one global rules for detectingdiscontinuities:
1. α for which the twice differenced series takes minimumvalue (following a non-negative one)
2. Values of α for which negative values persist afterrepeatedly smoothing twice differenced series by runningodd-spaced median smoothers (→ smooth)
3. α that minimizes BIC for splitting the series into two partsand fitting linear (quadratic) functions to each
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Order-α based outlier-detection oaoutlier
oaoutlier: Syntax
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Order-α based outlier-detection oaoutlier
Graph. output oaoutlier: (regional health production in Bavaria)
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Order-α based outlier-detection oaoutlier
oaoutlier: Saved Results
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References
References
Aigner, D., Lovell, C.A.K. & Schmidt, P. (1977). Formulation and estimation ofstochastic frontier production function models, Journal ofEconometrics 6: 21–37.
Aragon, Y., Daouia, A. & Thomas-Agnan, C. (2005). Nonparametric frontier estimation:a conditional quantile based approach, Econometric Theory21: 358–389.
Cazals, C., Florens, J.P. & Simar, L. (2002). Nonparametric frontier estimation: Arobust approach, Journal of Econometrics 106: 1–25.
Charnes, A., Cooper, W.W. & Rhodes, E. (1978). Measuring efficiency of decisionmaking units, European Journal of Operational Research 2: 429–444.
Daraio, C. & Simar, L. (2007). Advanced Robust and Nonparametric Methods inEfficiency Analysis: Methodology and Applications, Springer, NewYork.
Deprins, D., Simar, L. & Tulkens, H. (1984). Measuring laborefficiency in post offices,in: Marchand, M., Pestieau, P. and Tulkens, H. (eds.), The Performanceof Public Enterprises: Concepts and Measurement, Elsevier,Amsterdam, 243–267.
Ji, Y.B. & Lee, C. (2010). Data envelopment analysis, The Stata Journal 10, 267–280.
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