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Stochastic DEA: Myths and misconceptions Timo Kuosmanen (HSE & MTT) Andrew Johnson (Texas A&M University) Mika Kortelainen (University of Manchester) XI EWEPA 2009, Pisa, Italy

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2 What is stochastic DEA? DEA is truly a stochastic frontier estimation method, and it is incorrect to classify it as a deterministic method. Banker & Natarajan (2008) Operations Research, p.49

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3 What is stochastic DEA? Term stochastic (from Greek Στοχος for aim or guess) generally refers to statistical random variation

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4 Elements of random variation in DEA Random sampling of observations from the production possibility set (sampling error) Random sampling of observations outside the production possibility set (outliers) Random outcome of production process (stochastic technology) Random measurement errors, omitted variables, and other disturbances (stochastic noise)

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5 Common myths and misconceptions Confusing stochastic noise with sampling variation, outliers, or stochastic technology Statistical inference on sampling error is believed to improve robustness to noise Robustness to outliers is seen as the same as robustness to noise (or at least closely related)

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6 Sampling error True frontier input x output y

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7 Sampling error True frontier Random sample of observations (DMUs, firms) y x

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8 Sampling error True frontier Random sample of observations (DMUs, firms) y x

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9 Sampling error True frontier Random sample of observations (DMUs, firms) y x

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10 Sampling error True frontier DEA frontier y x

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11 Statistical foundation of DEA –Banker (1993) Management Science –Korostelev, Simar & Tsybakov (1995) Annals Stat. –Kneip, Park & Simar (1998) Econometric Theory –Simar & Wilson (2000) JPA Deterministic technology No outliers or noise Data randomly sampled from the PPS DEA frontier converges to the true frontier as the sample size approaches to infinity In a finite sample, DEA frontier is downward biased

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12 Statistical foundation of DEA Statistical inference on sampling error is possible by using –Asymptotic sampling distribution (Banker 1993) –Bootstrapping (Simar & Wilson 1998) Such inferences have nothing to do with –outliers –stochastic technology –stochastic noise

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13 Bootstrapping Purpose of the smooth consistent bootstrap (Simar & Wilson 1998, 2000) is to mimic the original random sampling to estimate the sampling bias Bias corrected DEA frontier will always lie above the original DEA frontier In noisy data, DEA tends to overestimate the frontier Assuming away noise, and correcting for the small sample bias by bootstrapping, we will shift the frontier upward => If noise is a problem, then bias correction will only make it worse

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14 Simulated example y x

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15 Simulated example y x

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16 Simulated example y x

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17 Critique of Löthgren & Tambour (LT) LT bootstrap involves measuring the distance from a different, random (as opposed to fixed) point to the [frontier] on each replication of the bootstrap Monte Carlo exercise. It seems entirely unclear what this procedure estimates. Certainly, it does not estimate anything of interest. … LT method assumes not only that [the frontier] is unknown, but also (implicitly) that the point from which one wishes to measure distance to the frontier is unknown. This is absurd. Simar & Wilson (2000), JPA, pp

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18 Outliers True frontier Outliers y x

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19 Outliers True frontier DEA frontier y x

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20 Outliers –Super-efficiency approach (Wilson 1995 JPA) –Peeling the onion; context dependent DEA (Seiford & Zhu 1999 Management Science) –Robust efficiency measures / efficiency depth (Kuosmanen & Post 1999 DP, Cherchye, Kuosmanen & Post 2000 DP) –Conditional order-m and order-α quantile frontiers (Aragon, Daouia & Thomas-Agnan 2002 DP; Cazals, Florens & Simar 2002 J Econometrics; Daouia & Simar 2007 J Econometrics; Daraio & Simar 2007 book) Deterministic technology Improve robustness to outliers by not enveloping the most extreme observations Outliers are different from noise –Noise affects all observations

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21 Stochastic technology Pr.[f(x)f]= 0.50 Pr.[f(x)f]= 0.05 Pr.[f(x)f]= 0.95 y x

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22 Stochastic technology y x Pr.[f(x)f]= 0.50 Pr.[f(x)f]= 0.05 Pr.[f(x)f]= 0.95

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23 Chance constrained DEA –Land, Lovell & Thore (1993) Managerial & Decision Econ. –Olesen & Petersen (1995) Management Science –Cooper, Huang & Li (1996) Annals of OR –Huang & Li (2001) JPA Stochastic technology, stochastic noise, both?

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24 Chance constrained stochastic DEA Huan & Li (2001) JPA Assume inputs and outputs are multivariate normal random variables, with known expected values and covariance matrix

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25 Chance constrained stochastic DEA How do we get knowledge about the expected values of inputs and outputs? –Cannot be estimated from cross-sectional data –Panel data estimation would require that the true inputs and outputs do not change over time How do we get knowledge about the variances and covariances of the error terms??? Uncertainty of the parameter estimates not taken into account in the model

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26 Stochastic noise True frontier y x

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27 Stochastic noise True frontier y x

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28 Stochastic noise True frontier y x

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29 Stochastic DEA models to deal with noise DEA+ –Gstach (1998) JPA –Banker & Natarajan (2008) Operations Research Stochastic DEA –Banker, Datar & Kemerer (1991) Management Science Stochastic FDH/DEA estimators –Simar & Zelenyuk (2008) DP. Stochastic Nonparametric Envelopment of Data (StoNED) –Kuosmanen (2006) DP; Kuosmanen & Kortelainen (2007) DP.

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30 Stochastic DEA models to deal with noise Estimation of a fully deterministic frontier based on data perturbed by noise –The shape of frontier can be estimated without parametric assumptions Estimation of inefficiency (efficiency scores) is very challenging in cross-sectional setting –Observed output contains the noise term –Only conditional expected value can be estimated –Even the SFA efficiency estimator is not consistent!

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31 Stochastic DEA models to deal with noise In cross-sectional setting, identifying inefficiency and noise requires some strong assumption –Assuming away noise completely is a strong assumption, too Distributional assumptions do not influence the efficiency rankings –Ondrich & Ruggiero 2001, EJOR

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32 Conclusions Stochastic noise should not be confused with sampling error, outliers, or stochastic technology Correcting for small sample bias by bootstrapping does not improve robustness to noise; it can even make things worse Improving robustness to outliers is different from stochastic noise that perturbs all observations

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