Upload
noel-patterson
View
215
Download
2
Tags:
Embed Size (px)
Citation preview
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