Stochastic Frontier Analysis of Efficiency of Moroccan Municipalities

Stochastic Frontier Analysis of Efficiency of Moroccan Municipalities Rachida El Mehdi Institut de Statistique,UCL Promoter: Professor Ch. Hafner E...
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Stochastic Frontier Analysis of Efficiency of Moroccan Municipalities

Rachida El Mehdi Institut de Statistique,UCL Promoter: Professor Ch. Hafner

Efficiency analysis •Basic idea –Comparison between the Decision Making Units DMU (firms, for example) in order to know how the inputs are used to produce outputs. • Nonparametric Data Envelopment Analysis (DEA) or the Free Disposal Hull (FDH), • Stochastic Parametric Frontier Analysis (FSA)



Stochastic Frontier Analysis (SFA) • Error term is divided in two independent terms,

ε i = vi − u i

i : Decision Making unit number i (DMUi) • vi reflects the pure randomness or the usual

statistic noise, v i ~ N (0, σ v2 ) • u i reflects the technical inefficiency, and u is a non-negative error term (ui ≥ 0.) u i = 0 for a technically efficient decision unit

Stochastic Frontier Analysis (SFA) • Some possible distribution of u i – Half-Normal u i ~ iidN + 0, σ u2 – Trancated Normal u i ~ iidN (µ , σ u2 ) truncated at zero – Exponentiel u i ~ iidΕxp 1  σu  – Gamma …

(

)

Stochastic Frontier Analysis (SFA) • Stochastic model (Production Function) with cross-sectional data

y i = f ( xi , β ) + vi − u i

Where

f ( xi , β )

:The production technology, it is assumed

either a Cobb-Douglas or a translog function (less restrictive).

yi : The observed output for observation

i (one logged output);

xi

: A vector of inputs for observation i (logged);

β

: A vector of parameters to be estimated;

Estimation of the stochastic model • Estimation can be made using – Maximum Likelihood (ML) method – Corrected Ordinary Least Squares (COLS) method. Greene (1980) proposes to correct the bias by shifting β 0 ,

βˆ * = βˆ + ε * 0

where

0

ε * = max(ε )

Estimation of the stochastic model • How to separate the error term into two components v and u – ε i can be estimated as εˆi = yi − f xi , βˆ

(

)

– Jondrow, Knox Lovell, Materov and Schmidt (1981) have proposed a decomposition by considering the expected value of u , conditional on ε = v − u

( ε)

Eu



Estimation of the stochastic model

The half-normal distribution (For example) h(u ) =

 1 2 exp− u  2 2π σ u  2σ u  2

u≥0

,

  1 2 1 2   g (v, u ) = f (v ).h(u ) = v − u  exp − 2 2 πσ vσ u   2σ v 2σ u  1

Replace

v =u +ε

to obtain

g (u, ε )

  1 2  (1 − Φ )exp − 2 ε  g (ε ) = ∫ g (u , ε )du = 0 2π σ    2σ  µ*   −  φ ( ) g u , ε  σ*  gu = u µ σ = + E ⇒ ε * * g (ε ) ε  µ  1 − Φ − *   σ*  +∞

( )

2

( )

Estimation of the stochastic model

( )

• The half-normal distribution (cont.) •

uˆ = E u ˆ ε

2 σ ε avec µ * = − u σ2

and

σ ≥0 λ= σ u

,

σ v2σ u2 σ = σ2 2 *

,

σ 2 = σ v2 + σ u2

is a measure of asymmetry

v

(Skewness) of the disturbance term

ε

.

Estimation of the stochastic model • The maximum likelihood estimator (MLE) of ϕ = (σ , λ , β ) is

ϕˆ ML = arg max l (ϕ ) ϕ and

Estimation of the stochastic model • Estimation methods • Analytical estimation (Not usually possible) Alternative: • Numerical estimation; • Monte Carlo Simulations.

Application • DMUs : 91 (1298) municipalities (DMUs) • One input : Recipe of functioning (Urban tax, tax on the collection of the waste, subsidies…) • One output : Financial autonomy

Estimation of the stochastic model • The half-normal distribution of u i (a and b) v i ~ iidN (0, σ v2 ) and u i ~ iidN + (0, σ u2 ) • The truncated-normal distribution of u i (c and d) vi ~ iidN (0, σ v2 ) and u i ~ iidN(µ,σ u2 ) truncated at zero •

u i ~ iidN (mi , σ u2 ) truncated at zero where mi = ziδ (e and f)

Results • N=91 : does not provides avalid TE i due to the positive Skewness of the distribution of

ε

Results

Results • Table1 and Fig.2 indicate • Efficiencies are different according to distribution of u i ; • Efficiencies are not more different according to Cobb-Douglas or translog functions; • The two models with the half-normal distribution (a and b) provide smaller technical efficiencies than the four others; • λ = 3.537≠0 , no problem of skewness.

Results

Include the Instrumental Variable in the estimation • When cov(x, ε ) ≠ 0 and E(v/ Z) = 0 Estimation in two Stages – x = Z ' Π +ν is estimated by OLS – y = xˆ ' β + ε is estimated by OLS (ML) βˆ

IV

(

= x PZ x '

)

−1

( )

x PZ y with PZ = Z Z Z '

'

−1

Z'

Include the Copula in the estimation • If U ⊥V it is recommended to find the joint density g(u, v) with copula Then or

G (u, v ) = Cθ (F1 (u ), F2 (v ))

g (u , v ) = f1 (u ). f 2 (v ).cθ (F1 (u ), F2 (v ))

Include the Copula in the estimation • Farlie-Gumbel-Morganstern (FGM) copula g θ (u , v ) = f1 (u ). f 2 (v ).cθ (F1 (u ), F2 (v )) = f1 (u ). f 2 (v ).[1 + θ − 2θF1 (u ) − 2θF2 (v ) + 4θF1 (u )F2 (v )]  2 = σ  u

 u   v  .φ    σ σ u    v

φ 

   

  2  u   v   − 2θΦ .1 + θ − 2θ  Φ  σ σ σ  u u    v  

 2  u   v   + 4θ   Φ  Φ    σ σ σ u u     v 

  .  

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