Efficiency of Public Spending in Developing Countries: A Stochastic Frontier Approach

William GreeneStern School of BusinessWorld Bank, May 23, 2005

Agenda

Theory for Stochastic Frontier Models

Aplication to IMF Health and Education Data

(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

The Production Frontier

Input

Output

A Textbook Definition of the ‘Production Function

Modeling The Production Frontier

Input

Output

Data Envelopment Analysis (LP) Approach

Modeling The Production Frontier

Input

Output

A Regression Approach

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

The Stochastic Frontier ‘Model’

There exists a production ‘function’ The data contain idiosyncratic noise

Measurement errorOmitted small effects

The theory of the production function applies to the specific firm – ‘the best’ is specific to the firm.

A Formal Model of Production

Technical and Allocative Inefficiency

An Econometric Model

( )y f x

( ) = 1.( )

y

TE y,f

xx

Parametric Frontier Model

( , ) = f TEy x

ln ln ( , ) + ln

= ln ( , ) -

= f TEy

f u

x

x

Technical Efficiency = Exp(-u)

Regression Based Frontier Model

ln ( [ ]) ( [ ])

* *.

i i i i i

i i

y E E

x

x

ln +

ln 0

[ ] 0

i ii

i i

i

= + y

TE

E

x

Estimate TEi

ln * [ ] i i i i ie y a u E ub x

exp( [ ])

exp( [ ])i ii

m mm

E uTE TE = E uTE TE

Corrected and Modified OLS

Stochastic Frontier

( ) iviii = fy eTEx

ln +

= + .

i i ii

i i

= + v uy

+

x

x

Statistical Model

Normally distributed ‘noise’ Inefficiency

Half normalOther kinds of positive random variables

(exponential, gamma, etc.)

The Normal-Half Normal Model

2

2

1( ) [0, ] ,

| |

1( ) [0, ] ,

ii v i

v v

i i

ii u i

u u

vf v Normal v

and

u U

Uf U Normal U

Underlying Density2

2 22 2 2 2

( / )2( ) exp

2( )2 ( )i u v i

iu vu v u v

f

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

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

Evidence of Inefficiency

Decomposing ε

2

2 2

[( 2) / ][ ]

[ ] [( 2) / ]u

u v

Var u

Var

Useful Formulation2 2,

2( )

uu v

v

i iif

2N

i=1

- 1Log , , ) = - ln - constant + ln -

2i i-L( , N

lni ii = y x

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

Multiple Outputs - Costs

121 1 1

121 1 1

1 1

Stochastic Cost Frontier

ln ln ln ln (input prices)

ln ln ln (outputs)

ln ln

K K K

i k kl k lk k l

M M M

m mr m rm m r

M K

mk m k i im k

C w w w

y y y

y w v u

Technical and Allocative Inefficiency

Any suboptimal decision must increase costs – ‘efficient’ costs are minimum.

Multiple Outputs - DistanceOutput 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

Output Distance Stochastic Frontier

0 = lny1 +

lnDO(x, y2/y1, y3/y1,...,yM/y1) +

v + ln[exp(u)]

-lny1 = lnDO(x, y2/y1, y3/y1,...,yM/y1) + v + u

Measuring Inefficiency

i 2

( ) -| = + where =

1+ ( )i i

i i ii

E u

TˆObservation is = y -

How do we decompose this into two parts?

Jondrow, Materov, Lovell, Schmidt result:

x = v - u

Measurement and Estimation

Cross SectionsProduction parametersMeasured heterogeneity (In)efficiency

Panel DataUnmeasured inefficiency Is inefficiency constant across time?Other forms of heterogeneity

World Bank Data

HealthOutputs: Life expectancy, immunization Inputs: Public and private spending, literacy

EducationOutputs: Enrollment, literacy, completion,

years of schooling Inputs: Teachers, adult literacy, spending

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)

Health Outcomes Model

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

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

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

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

Country RanksCountry 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

Comparing Efficiencies

Immunizations

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

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

Inefficiencies from Panel Model

Obs. Country Inefficiency Obs. Country Inefficiency1 6 0.0179612 2 10 0.02592383 16 0.262076 4 18 0.07898085 20 0.132446 6 21 0.05178967 22 0.109779 8 27 0.04849119 29 0.159224 10 30 0.036099711 31 0.0938481 12 34 0.47113313 35 0.275324 14 40 0.084177715 41 0.214609 16 42 0.14769517 43 0.0681757 18 45 0.085683219 46 0.132356 20 47 0.0659919

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

Efficiencies from Distance Function

Analyzing Distance InefficiencyLinear 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

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

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