8 Efficiency Stochastic Frontiers: a Panel Data
Analysis for Spanish Airports (1992-1994)
Pablo Coto-Milln
Department of Economics
University of Cantabria (Spain)
Gema Carrera-Gmez
Department of Economics
University of Cantabria (Spain)
Juan Castanedo-Galn
Department of Transports
University of Cantabria (Spain)
Miguel A. Pesquera
Department of Transports
University of Cantabria (Spain)
Vicente Inglada
Department of Economics
University Carlos III of Madrid (Spain)
Rubn Sainz
Department of Economics
University of Cantabria (Spain)
Ramn Nez-Snchez
Department of Economics
University of Cantabria (Spain)
8.1 Introduction
In this work we analyse and estimate the economic efficiency of a sample of 33
Spanish civil airports in the 1992-1994 period. With this aim, we have estimated a
122 P. Coto-Milln et al.
frontier-cost function by applying the panel data technology, which enables us to
c1assify the different Spanish airports under their economic efficiency.
A frontier cost function represents the minimum cost at which a particular level of
output is produced given the technology and the prices of the production factors
used. The basic specification of a cost frontier is:
Ch = f (w, y; E) exp (Uh), uh, t 0 (8.1)
where C is the cost of the h-th firm, w is the price vector of the inputs, f (w, y; E)
represents the minimum cost and uh represents the deviations of the cost effec-
tively achieved by each firm with respect to the minimum cost. Such deviations
will be above the cost frontier rather than below it.
A cost frontier can be obtained through the estimation of deterministic or sto-
chastic frontiers. The former assume that the deviations towards the frontier are
exc1usively due to an economic inefficiency. That is to say, the economic effi-
ciency is defined as (Aigner and Chu 1968):
EE = C / f (w, y; E) = exp (uh) (8.2)
On the other hand, the stochastic frontiers take into account the random and un-
controlled factors which may affect the production and costs of a firm - i.e., envi-
ronmental and weather conditions, problems in the supply of the productive fac-
tors, etc. or measurement errors - (Aigner et al. 1997). Therefore, the error term
falls now into two categories:
C = f (w,y; E) exp (eh), eh = vh+uh (8.3)
where vh accounts for the random effect and uh accounts for the economic ineffi-
ciency.
The cost frontiers may be parametric (they may impose a particular functional
form) or non-parametric. In this research we have used a data panel to estimate the
translogarithmic parametric function, for which we have employed an econometric
model.
8.2 The Model
With the aim of obtaining the economic efficiency of the different civil airports,
we estimate a frontier-cost function. In order to put this model into practice it is
necessary to choose a particular functional form. However, this confers a series of
characteristics on the technology studied without accurate knowledge of the cer-
tainty of such properties. Therefore, it is important to choose flexible functional
forms which place the least number of restrictions on the technology. Conse-
quently, we have chosen a flexible functional form: a multiproduct translogarith-
mic function (Translog). Then, the cost function is specified as follows:
8 Efficiency Stochastic Frontiers: a Panel Data Analysis for Spanish Airports 123
m
1m m n
CVh A D r ln y rht rs rht sht
D ln y ln y Ei ln w iht
r 1 2r 1 s 1 i 1
n n m n
1
2i1 j1
Eij ln w iht ln w jht Uri ln y rht ln w iht It t
r 1 i 1
(8.4)
m n
1
Iu t 2 Urt ln y rht t Eit ln w iht t H ht
2 r 1 i 1
where i, j = 1...n is the price of the different inputs; r, s =1...m is the number of
outputs; h = 1...H is the number of civil airports, A is the constant, t is a time trend
and Hht is the random disturbance term. On this function, we have imposed the
homogeneity conditions of degree one on the factor prices and the symmetry con-
ditions.
As seen earlier, assume that Hht has two components: Hht = vht + uh, where vht ac-
counts for the random disturbance of the usual characteristics [iid, N (0, VA)] and
uh (> O) is supposed to capture the inefficiency degree of the h-th firm. This error
component follows an unknown distribution iid, D (P, VB).
Schmidt and Sickles (1984) reformulate the model as follows:
A* = A + P (8.5)
and
uh* = uh - P (8.6)
Therefore, uh* is iid with E(uh) = 0. In this sense, both errors have zero as average
value, therefore the standard panel data models can be applied (fixed effect model
or random effect model). The choice of either model will depend on whether uh is
correlated or not with the explanatory variables recorded:
- The fixed effect model assumes that the individual effects are specific constants
for each firm, so that they would take part in the translog functional form.
- The random effect model assumes that the individual effects follow an un-
known distribution and therefore would take part in the random disturbance
term.
If we assume that the above-mentioned correlation exists, the random effect model
would be inconsistent since the explanatory variables would be correlated with the
part of random disturbance in correspondence with the individual effect, so that
the model of fixed effects must be applied since it lacks this problem and, there-
fore, the estimators are consistent. On the other hand, if we assume that there is no
such correlation, the application of the random effect model is also consistent and
more efficient.
With the aim of determining which of these models is suitable in our case, we
have applied the Hausman test which identifies any correlation between the fixed
effects and the exogenous variables. Therefore, to account for the individual effect
(uh) we have introduced a different dummy variable for each airport in the cost
124 P. Coto-Milln et al.
function. This individual effect would be the economic inefficiency. In this sense,
the estimated function is a stochastic frontier which isolates the random effects
(represented by the error term) thus called economic inefficiency.
8.3 The Data
The data used in the estimation of the cost function have been obtained from a
panel of 33 civil airports of national interest observed during the 1992-1994 pe-
riod. The variable which depends on the model (CV) is the total variable cost -
equal to the sum of all the costs: employee costs, depreciation and intermediate
consumptions -. The model inc1udes a single production variable obtained when
aggregating the airport activity, the total of passengers moved in the airport (the
passengers embarked and disembarked). The model also inc1udes three variable
inputs: labour (L), capital (K) and intermediate consumptions (E). Prices are ob-
tained as follows: the price of labour (wL) is the quotient obtained by dividing the
whole of the employee costs by the total number of workers employed; the price
of capital (wK) is obtained dividing the amortization of the period by the number
of linear meters of the quays; and the price of the intermediate consumption (wE)
is obtained as the quotient of the consumption, external supplies and services costs
and other expenses divided by the airport activity measured in passengers.
Table 8.1 shows the statistical aspects of inputs and outputs.
Table 8.1. Summary statistics
Passengers Labour Capitala I. Consumpt.
(thousands) (Number) (pesetas) (pesetas)
1992
Maximum 18,096 818 54,368,078 3,506,631
Minimum 100 37 569,369 23,846
Average 785 142 4,715,669 174,053
Standard Dev. 4,094 183 11,125,006 762,786
1993
Maximum 17,339 823 60,498,612 3,910,108
Minimum 91 37 650,799 24,869
Average 774 143 5,132,705 195,017
Standard Dev. 4,056 185 12,359,387 830,959
1994
Maximum 17,786 780 70,650,913 4,589,083
Minimum 93 37 759,989 25,351
Average 786 139 5,991,764 209,403
Standard Dev. 4,173 181 14,431,289 964,753
a
Aproximated by the amortization estimated in constant pesetas
Source: Memorias Anuales de Aeropuertos Espaoles y AENA.
8 Efficiency Stochastic Frontiers: a Panel Data Analysis for Spanish Airports 125
8.4 Econometric Results
Table 8.2 shows the economic efficiency indexes and returns to scale for each
civil airport.
Table 8.2. Economic efficiency indexes and returns to scale
Civil Airports Economic Efficiency Indexes Returns to Scale
1-Mlaga 0.997 <1
2-Las Palmas 0.617 >1
3-Granada 0.468 >1
4-Santiago 0.693 >1
5-Madrid 1 =1
6-Asturias 0.534 >1
7-Bilbao 0.708 >1
8-Vigo 0.547 >1
9-Almera 0.694 >1
10-Alicante 0.937 <1
11-Pamplona 0.470 >1
12-Melilla 0.423 >1
13-San Sebastin 0.390 >1
14-Santander 0.448 >1
15-Barcelona 1 =1
16-Vigo 0.544 >1
17-Zaragoza 0.279 >1
18-Valladolid 0.424 >1
19-Reus 0.589 >1
20-Murcia 0.484 >1
21-Jerez de la Frontera 0.603 >1
22-Ibiza 0.800 >1
23-Valencia 0.681 >1
24-Fuerteventura 0.785 >1
25-Gran Canaria 0.990 >1
26-Hierro 0.295 >1
27-Girona 0.528 >1
28-Tenerife Sur 0.982 >1
29-Tenerife Norte 0.658 >1
30-La Corua 0.576 >1
31-Palma de Mallorca 1 >1
32-Sevilla 0.631 >1
33-Menorca 0.692 >1
126 P. Coto-Milln et al.
8.5 Conclusions
The most efficient airports are the ones of Madrid, Barcelona, Palma de Mallorca,
Malaga, Gran Canaria, Tenerife South, Alicante, Ibiza, Fuerteventura, Menorca
and Bilbao.
These airports have the highest traffic of passengers (between 3 million and 19
million).
Airports of medium efficiency are the ones of Tenerife North, Valencia, Seville,
Santiago, Almeria, Las Palmas, Asturias, Vigo, Reus, Jerez de la Frontera, Girona
and La Corua. These airports have medium volumes of traffic (between 400
thousand and 3 million).
The airports of Granada, Pamplona, Melilla, San Sebastian, Santander,
Zaragoza, Valladolid, Murcia and Hierro have a low efficiency, with volumes of
traffic of up to 400 thousand passengers. The airports of Madrid and Barcelona
present constant returns to scale because they have extinguished their scale
economies as they have achieved their optimum size.
However, the airports of Malaga and Alicante present decreasing returns. A
possible reason for this is the strong stationarity in traffics, which requires that
airports are big in order to answer to the peaks in demand, while traffics are low
the rest of the year.
The remaining airports present increasing returns, as expected. These airports
will extinguish their returns to scale economies as they increase the traffic (we
have estimated an average increase around 5% during the last 10 years), thus
achieving the optimum size.
References
Aigner DJ, Chu SF (1968) On estimating the industry production function. American
Economic Review 58: 226-239.
Aigner DJ, Lovell CK, Schmidt P (1997) Formulation and estimation of stochastic frontier
production function models. Journal of Econometrics 6: 21-37.
Memorias Anuales de los Aeropuertos Espaoles (1992-1994). Direccin General de Aero-
puertos, AENA.
Schmidt P, Sickles RC (1984) Production frontiers and panel data. Journal of Business and
Economic Statistics 2: 367-374.
