1. Efficiency of Public Spending in Developing Countries: A Stochastic Frontier Approach William Greene Stern School of Business World Bank, May 23, 2005

2. Agenda Theory for Stochastic Frontier Models Aplication to IMF Health and Education Data

3. (In)Efficiency Production and Efficiency in Production What do we mean by ‘inefficiency?’  Economically  Mathematically – in the Model Measurement  Relative: Who is doing it ‘well?’  Absolute: Benchmarks

4. The Production Frontier Input Output A Textbook Definition of the ‘Production Function

5. Modeling The Production Frontier Input Output Data Envelopment Analysis (LP) Approach

6. Modeling The Production Frontier Input Output A Regression Approach

7. Questionable Assumptions Implication that some agents in the ‘sample’ are perfectly efficient Assumption that the measured data reflect only the underlying process and production inefficiency

8. The Stochastic Frontier ‘Model’ There exists a production ‘function’ The data contain idiosyncratic noise  Measurement error  Omitted small effects The theory of the production function applies to the specific firm – ‘the best’ is specific to the firm.

9. A Formal Model of Production

10. Technical and Allocative Inefficiency

11. An Econometric Model

12. Parametric Frontier Model Technical Efficiency = Exp(-u)

13. Regression Based Frontier Model

14. Estimate TE i

15. Corrected and Modified OLS

16. Stochastic Frontier

17. Statistical Model Normally distributed ‘noise’ Inefficiency  Half normal  Other kinds of positive random variables (exponential, gamma, etc.)

18. The Normal-Half Normal Model

19. Underlying Density

20. Inefficiency in the Disturbance OLS estimates the model parameters consistently – save for the constant term Residuals contain information about inefficiency. Skewness does not require a consistent estimate of the constant term

21. Log Life Expectancy at Birth +----------------------------------------------------+ | Ordinary least squares regression | | LHS=LOGBIRTH Mean = 4.122497 | | Standard deviation =.1985908 | | Residuals Sum of squares = 6.896716 | | Standard error of e =.1477330 | | Fit R-squared =.4518070 | +----------------------------------------------------+ +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | +---------+--------------+----------------+--------+---------+ Constant 3.12834614.07302871 42.837.0000 LOGAID -.00522511.00454926 -1.149.2516 LHPUB.08459371.00935802 9.040.0000 LHPRIV.02664818.01185472 2.248.0253 Skewness measure of Residuals = -1.0471

22. Evidence of Inefficiency

23. Decomposing ε

24. Useful Formulation

25. Interesting Extensions Heteroscedasticity Heterogeneity  Variables that shift the production function  Variables that directly impact (in)efficiency  Unmeasured heterogeneity – cross country Other distributions than half-normal Analyzing Costs – Measures ‘economic’ inefficiency, both technical and allocative

26. Multiple Outputs - Costs Technical and Allocative Inefficiency Any suboptimal decision must increase costs – ‘efficient’ costs are minimum.

27. Multiple Outputs - Distance Output Distance: DO(x,y) = Min( : y/ is producible with x) Output distance is < 1. Y1 DO(x, y2/y1, y3/y1,...,yM/y1) TO = 1

28. Output Distance Stochastic Frontier 0 = lny 1 + lnDO(x, y 2 /y 1, y 3 /y 1,...,y M /y 1 ) + v + ln[exp(u)] -lny 1 = lnDO(x, y2/y1, y3/y1,...,yM/y1) + v + u

29. Measuring Inefficiency

30. Measurement and Estimation Cross Sections  Production parameters  Measured heterogeneity  (In)efficiency Panel Data  Unmeasured inefficiency  Is inefficiency constant across time?  Other forms of heterogeneity

31. World Bank Data Health  Outputs: Life expectancy, immunization  Inputs: Public and private spending, literacy Education  Outputs: Enrollment, literacy, completion, years of schooling  Inputs: Teachers, adult literacy, spending

32. Sample Data 232 countries and political units ‘Panel’ 1975-2002 Sparse: Most observations post 1996 Missing data throughout will inhibit panel data treatments Countries restricted to those analyzed by Herrera and Pang Years restricted to 1996-2002 (as per H&P) (Results will not identify specific countries)

33. Health Outcomes Model lnHealth =  o +  1 LitAdult +  2 logAidRev +  3 HIV/AIDS +  4 logHPublic +  5 logHPrivate + v – u.

34. Life Expectancy at Birth +---------------------------------------------+ | Dependent variable LOGBIRTH | | Number of observations 243 | | Sigma(v) =.06387 | | Sigma(u) =.11149 | | Sigma = Sqr[(s^2(u)+s^2(v)]=.12850 | | Stochastic Production Frontier, e=v-u. | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | +---------+--------------+----------------+--------+---------+ Primary Index Equation for Model Constant 3.57745980.06075876 58.880.0000 LITADULT.00244588.00038204 6.402.0000 LOGAID.00054422.00301162.181.8566 DUM_AIDS -.23005492.01691262 -13.603.0000 LHPUB.01113797.00795436 1.400.1614 LHPRIV.04558010.00921797 4.945.0000 Variance parameters for compound error Lambda 1.74554534.25629061 6.811.0000 Sigma.12849508.00048314 265.960.0000

35. DPT Immunizations +---------------------------------------------+ | Dependent variable LOGDPT | | Number of observations 469 | | Log likelihood function 87.30556 | | Sigma(v) =.03097 | | Sigma(u) =.37963 | | Sigma = Sqr[(s^2(u)+s^2(v)]=.38089 | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Primary Index Equation for Model Constant 3.91762187.08498639 46.097.0000 LITADULT.00263122.00038971 6.752.0000 76.6400192 LOGAID -.394773D-04.00385818 -.010.9918 -.07081737 DUM_AIDS -.01928035.01436402 -1.342.1795.28784648 LHPUB.01986881.00995082 1.997.0459 8.90031837 LHPRIV.03622585.01102776 3.285.0010 8.71678629 Variance parameters for compound error Lambda 12.2596748 2.39146775 5.126.0000 Sigma.38089048.00067166 567.091.0000

36. Estimated Efficiencies Estimated Efficiency: Year 2000 Values, Four Health Outcomes Line Observ. COUNTRY EFFLIFE EFFDALE EFFMEA EFFDPT 1 166 6.96328.92603.93791.95085 2 250 9.90777.87812.83027.83352 3 278 10.94430.93589.85802.86545 4 446 16.91063.88950.95425.91250 5 502 18.97469.96516.86163.95570 6 558 20.95399.94191.90921.95931 7 586 21.94225.92957.83027.87057 8 614 22.93424.90825.88083.87891 9 698 25.92658.91586.92477.93395 10 754 27.95671.92049.92914.86042

37. Country Ranks Country Ranks for Computed Efficiency Measures, Sorted by Rank for Life Expectancy. Line Observ. COUNTRY RANKLIFE RANKDALE RANKMEA RANKDPT 1 166 6 1 1 64 14 2 250 9 2 2 42 94 3 278 10 3 3 85 93 4 446 16 4 9 1 50 5 502 18 5 7 22 92 6 558 20 6 11 35 44 7 586 21 7 6 57 55 8 614 22 8 10 50 69 9 698 25 9 5 10 57 10 754 27 10 8 27 85

38. Comparing Efficiencies

39. Immunizations

40. Rank Correlations Rank Correlation: Efficiency Measures, LIFE, DALE =.925 Rank Correlation: Efficiency Measures, LIFE, MEA =.288 Rank Correlation: Efficiency Measures, LIFE, DPT =.377 Rank Correlation: Efficiency Measures, DALE, MEA =.308 Rank Correlation: Efficiency Measures, DALE, DPT =.392 Rank Correlation: Efficiency Measures, MEA,DPT =.736

41. Panel Data Estimator +---------------------------------------------+ | Dependent variable LOGBIRTH | | Number of observations 183 | | Log likelihood function 227.3098 | +---------------------------------------------+ | Frontier model estimated with PANEL data. | | Estimation based on 82 individuals. | | Sigma(v) =.03705 |.06387 | Sigma(u) =.19172 |.11149 | Sigma = Sqr[(s^2(u)+s^2(v)]=.19526 |.12850 | Stochastic Production Frontier, e=v-u. | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | +---------+--------------+----------------+--------+---------+ Primary Index Equation for Model Constant 3.71326271.11495348 32.302.0000 LOGAID.00071987.00775194.093.9260 LHPUB.02323586.01399616 1.660.0969 LHPRIV.04459252.01963929 2.271.0232 DUM_AIDS -.16684671.02147093 -7.771.0000 Variance parameters for compound error Lambda 5.17488982 1.45616953 3.554.0004 1.7455 Sigma(u).19171609.01999430 9.589.0000.12899

42. Inefficiencies from Panel Model Obs. Country Inefficiency 160.01796122100.0259238 3160.2620764180.0789808 5200.1324466210.0517896 7220.1097798270.0484911 9290.159224 10300.0360997 11310.0938481 12340.471133 13350.275324 14400.0841777 15410.214609 16420.147695 17430.0681757 18450.0856832 19460.132356 20470.0659919

43. Distance Function +---------------------------------------------+ | Number of observations 127 | | Sigma(v) =.06214 | | Sigma(u) =.09900 | | Sigma = Sqr[(s^2(u)+s^2(v)]=.11689 | | Stochastic Cost Frontier, e=v+u. | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Primary Index Equation for Model Constant 3.65109400.09657639 37.805.0000 DY2.04669894.07127361.655.5123.26693392 DY3 -.26698703.08520247 -3.134.0017.27715934 DY4.60557844.19916980 3.041.0024 -.15911336 LITADULT.00310847.00065067 4.777.0000 78.9670450 AIDREV.00049076.00146290.335.7373 2.25076999 LHPUB.00298839.01137482.263.7928 8.97024885 LHPRIV.03753580.01193765 3.144.0017 8.74912626 DUM_AIDS -.24810324.02169494 -11.436.0000.31496063 Variance parameters for compound error Lambda 1.59313299.31985029 4.981.0000 Sigma.11688816.00068853 169.765.0000

44. Efficiencies from Distance Function

45. Analyzing Distance Inefficiency Linear Regression of Distance (Multiple Outpot) Efficiency on Covariates +----------------------------------------------------+ | Ordinary least squares regression | | LHS=EFFDSTNC Mean =.9680386 | | Standard deviation =.1756748E-01 | | WTS=none Number of observs. = 61 | | Model size Parameters = 8 | | Degrees of freedom = 53 | | Residuals Sum of squares =.1140125E-01 | | Standard error of e =.1466690E-01 | | Fit R-squared =.3842809 | | Model test F[ 7, 53] (prob) = 4.73 (.0004) | | Info criter. LogAmemiya Prd. Crt. = -8.321091 | | Akaike Info. Criter. = -8.322611 | +----------------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |t-ratio |P[|T|>t] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Constant.98032770.03649461 26.862.0000 LOGPOPU.00713042.00939002.759.4510 3.99474313 LOGGDP.00090310.00395778.228.8204 8.44788653 LOGGOV -.00483267.00787183 -.614.5419 3.25658249 GINI -.04591459.03169753 -1.449.1534.41134873 LOGWAGE.00075284.00441985.170.8654 2.85301416 LOGPUBTO -.00334868.00799979 -.419.6772 4.07654175 DUM_AIDS -.01734261.01200003 -1.445.1543.13114754

46. Rankings of Distance Efficiencies Line Observ. Country EFFDSTNC RANK 1 166 34.97208 1 2 250 231.97138 2 3 278 154.95784 3 4 446 147.94805 4 5 502 129.94425 5 6 558 108.94150 6 7 586 100.94077 7 8 614 16.93006 8 9 698 140.92658 9 10 754 186.92587 10

47. Education Frontier +---------------------------------------------+ | Dependent variable LOGPSENR | | Number of observations 239 | | Akaike IC= -141.140 Bayes IC= -116.805 | | Sigma(v) =.10902 | | Sigma(u) =.23585 | | Sigma = Sqr[(s^2(u)+s^2(v)]=.25982 | +---------------------------------------------+ +---------+--------------+----------------+--------+---------+----------+ |Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+ Primary Index Equation for Model Constant 2.55223730.34206895 7.461.0000 LOGEDU.02245883.01543313 1.455.1456 4.79099195 LOGLITA.37967218.05205007 7.294.0000 4.26891824 LOGTCHR -.13990380.04471329 -3.129.0018 -3.37361566 LOGAID.00852680.00636758 1.339.1805 -.02032254 Variance parameters for compound error Lambda 2.16335588.35748068 6.052.0000 Sigma.25982370.00087845 295.775.0000