19. Statistical Approaches to. Data Variations Tuomas Koivunen S ysteemianalyysin. Laboratorio. Optimointiopin seminaari - Syksy 2007

Transkriptio

1 19. Statistical Approaches to Data Variations Tuomas Koivunen

2 Contents 1. Production Function 2. Stochastic Frontier Regressions 3. Example: Study of Texas Schools 4. Example Continued: Simulation Experiment 5. Summary 6. Home Assignment

3 1. Production Function

4 Production Function Definition and Use with DEA Definition of production function: How outputs are got from inputs in mathematical terms Of all the feasible combinations, only those that give maximum output for a specified set of inputs constitute the production function The link between DEA and production functions is that the deviations from a production function can be regarded as stochastic variations in technical efficiency Therefore, this is presented under the chapter Data Variations in the Book

5 2. Stochastic Frontier Regressions

6 Stochastic Frontier Regressions Composed Error We use a statistical regression model to give the output as a function of inputs: y = f (x ) +ε where ε represents random errors that occur in the dependent variable y Thus, the components of x are assumed to be known without error

7 Stochastic Frontier Regressions Composed Error Now, the random error ε is replaced by a 2- component term (therefore composed error ): ε = υ τ υ represents the random error component which may be positive, negative or zero τ is restricted to nonnegative ranges that represent values of y which fail to to achieve the efficient frontier. It is also assumed to have statistical distributions such as exponential

8 Stochastic Frontier Regressions The estimates of technical efficiency The estimates of technical efficiency are obtained from: ˆ τ = µ τ σ f F *( µ τ *( µ τ / σ ) / σ ) µ τ σ = τ 2 υ τ σ = σ υ + σ 2 τ σ υ + σ τ where and σ σ and where f* and F* represent the standard normal density and cumulative normal distribution functions with mean µ and variance σ 2

9 Stochastic Frontier Regressions The estimates of technical efficiency The efficiency corresponding to specified values for the components x are then estimated from: 0 ˆ e τ 1 Which is equal to one when τ^ = 0 and becomes 0 as τ^

10 Stochastic Frontier Regressions Log-linear production function To see how this measure of efficiency is used, we employ the log-linear Cobb-Douglas production function y = = β β 0 0 x x β 1 1 β 1 1 x x β 2 β e e ε υ τ so that ˆ = τ β 1 β 2 y = ye β 0 x 1 x 2 e υ The inefficiency is given as an output shortfall

11 3. Example: Study of Texas Schools

12 An example The Study Stochastic frontier regressions can be seen as competing with DEA but the two can also be combined as a study shows: Evaluating public schools in Texas with two inputs and one output It was found that evaluating the schools with the original form of the Cobb-Douglas production function, ln y ln β + β ln x ˆ + β ln x + = ˆ gave unsatisfactionary results ε

13 An example The Study Satisfactory results were, however, found when using a two-stage approach: In the 1st stage, schools were evaluated with DEA. In the second stage, the Cobb-Douglas was extended to incorporate the results of the first stage in the form of dummy variables An efficient school was given the Dummy value 1 An inefficient school was given the Dummy value 0 Production function became: ln y ln β + β ln x ˆ + β ln x ˆ + δ D + δ D ln x ˆ + δ D ln x + = ˆ ε

14 4. Example continued: Simulation Experiment

15 Simulation Experiment Motivation The study was followed by a simulation to examine the validity of the method with a revised set of data The Cobb-Douglas took the following form: y x x 55 = 0.75* e ε where the parameter values were known from the study: β β β = = =

16 Simulation Experiment Procedure 1) The random term e ε is used to generate random values i. The new y values containing these random terms are generated this way 2) The input values x 1 and x 2 are generated randomly as a bias avoiding mechanism i. These are inserted in the Cobb-Douglas function to provide the truly efficient values of y 3) The values for y^ are are calculated

17 Simulation Experiment Procedure 4) The inputs are adjusted to new values: x ˆ x ˆ τ 1 2 1, = = τ 2 x 1 x e 2 τ e 1 τ 0 2 where τ 1 and τ 2 represent input-specific technical inefficiencies drawn at random Against the stochastic frontier assumption, inefficiencies are impounded in the inputs Nevertheless, this reproduces a situation where the outputs tend to be too low for the inputs

18 Simulation Experiment Procedure 5) A subset of actual observations x 1 and x 2 is chosen at random to Determine if the first stage DEA identified the efficient DMUs Examine the effects of this efficient subset on the derived statistical estimates 6) With this new set of data with random variables, the parameter values and thus the efficient frontier was estimated Both Ordinary Least Square (OLS) estimates as well as Stochastic Frontier (SF) estimates were used

19 Simulation Experiment Results without Dummy Variables OLS: Case A Case B Case C Case D σ ε ² = 0.04 σ ε ² = σ ε ² = 0.01 σ ε ² = (1) (2) (3) (4) β 0 1,30 1,58 1,40 1,43 β 1 0,46 0,43 0,45 0,46 β 2 0,48 0,47 0,47 0,46 As can be seen, all parameter estimates are wide of the true values

20 An example Results without Dummy Variables SF: Case A Case B Case C Case D σ ε ² = 0.04 σ ε ² = σ ε ² = 0.01 σ ε ² = (1) (2) (3) (4) β 0 1,42 1,62 1,25 1,28 β 1 0,46 0,43 0,48 0,46 β 2 0,48 0,47 0,48 0,47 σ τ 0,15 0,11 0,15 0,15 σ ν 0,15 0,13 0,08 0,04 The estimates are quite close to the OLS estimates and thus wide from the true values as well

21 An example Two-stage Approach Results OLS: Case A Case B Case C Case D σε² = 0.04 σε² = σε² = 0.01 σε² = (1) (2) (3) (4) β0 1,07 1,47 1,28 1,34 β 1 0,49 0,43 0,46 0,47 β2 0,48 0,48 0,48 0,46 δ -1,57-2,30-1,50-1,50 δ1 0,16 0,26 0,16 0,16 δ 2 0,12 0,12 0,1 0,09 β1+dδ1 0,65 0,69 0,62 0,63 β2+dδ2 0,60 0,60 0,58 0,55 For the efficient DMUs with D=1 the estimates do not differ significantly from the true ones

22 An example Two-stage Approach Results SF: Case A Case B Case C Case D σ ε ² = 0.04 σ ε ² = σ ε ² = 0.01 σ ε ² = (1) (2) (3) (4) β 0 1,18 1,50 0,80 1,40 β 1 0,50 0,44 0,53 0,49 β 2 0,48 0,49 0,50 0,47 δ -1,60-2,40-1,25-1,55 δ 1 0,16 0,26 0,13 0,15 δ 2 0,11 0,13 0,09 0,09 σ ν 0,13 0,09 0,05 0,04 β 1 +Dδ 1 0,66 0,70 0,66 0,64 β 2 +Dδ 2 0,59 0,62 0,59 0,56 The estimates for efficient DMUs are good as with OLS

23 5. Summary

24 Summary The Key Takeaways Production functions give the output(s) as a mathematical function of the inputs DEA can give an estimator of a production function As an approach, stochastic frontier regressions have similarities with DEA: The deviations from the production function are given as a composed error with a random part and a technical inefficiency part DEA and stochastic frontier regressions can be combined as the example of Texas public schools shows

25 Summary Further Reading To understand deeper the links between stochastic frontier regressions and DEA, the following literature would be useful: D.J. Aigner, C.A.K. Lovell and P. Schmidt (1977), Formulation and Estimation of Stochastic Frontier Production Models, Journal of Econometrics 6, pp W. Meeusen, and J. Van den Broeck (1977) Efficiency Estimation from Cobb-Douglas Functions with Composed Error, International Economic Review 18, pp V. Arnold, I.R. Bardhan, W.W. Cooper and S.C. Kumbhakar (1994), New Uses of DEA and Statistical regressions for Efficiency Evaluation and Estimation - With an Illustrative Application to Public Secondary Schools in Texas, Annals of Operations Research 66, pp

26 6. Home Assignment

27 Home Assignment Finnish Speaking Students Tuottavuus Suomen yliopistoissa Tutustu raportin T. Räty, J. Kivistö, "Mitattavissa oleva tuottavuus Suomen yliopistoissa" pääpiirteisiin Perehdy erityisesti luvussa 3.1 (s ) tarkasteltuihin yliopistojen "laatua" kuvastaviin indikaattoreihin. Esitä noin yhdellä sivulla oma, mahdollisesti eri muuttujiin perustuva indikaattori, ja arvioi sen etuja ja puutteita. Ottaako esittämäsi indikaattori yliopiston koon tehokkuusnäkökulmasta tarkoituksenmuksella tavalla huomioon? Kotitehtävä tehdään yksilötyönä

28 Home Assignment German Speaking Students Hospital Capacity Planning Please write a one-page critical evaluation on the following report and describe its essential messages to the rest of the class on the 14 th of November (or later, if you cannot attend on that date): nalyse_krankenhausplanung.pdf You may collaborate