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
.