Economics and rural development Vol. 1, No 1, 2005 ISSN 18223346
23
ESTIMATING ECONOMETRIC MODEL OF AVERAGE TOTAL MILK
COST: A SUPPORT VECTOR MACHINE REGRESSION APPROACH
Reet Põldaru, Jüri Roots, AntsHannes Viira
Institute of Informatics, Estonian Agricultural University, Estonia
This paper gives an overview of the basic ideas underlying support vector machines (SVM) for regression and function estima
tion. A summary of currently used algorithms for training SVM is be presented. Application of SVM regression for estimating pa
rameters of econometric model of average total milk cost in Estonian farms is considered and possibilities of application of SVM re
gression in rural areas are discussed. Studies on implementation of SVM regression methods (algorithms) and software packages in
agricultural research and business must be extended.
Key words: econometric models, data mining, support vector machine regression, average total milk cost.
Introduction
In recent years the longterm prospects for the agri
cultural world markets have been subject of intensive dis
cussions mainly for two reasons:
• The rising concern is given to the food security
situation in a number of developing countries.
• The high level of support for the agricultural sec
tors and large production surpluses in many developed
countries, which could be exported only with the exten
sive use of export subsidies, result in the need for interna
tional efforts to liberalise the markets.
Through last decades, the use of economic models in
relation to agricultural policy issues has increased sub
stantially, and a big number of literature sources on these
issues is available. A number of different modelling ap
proaches have been applied.
Estonia is one of the new members of the European
Union. The EU enlargement brings for East European
countries a lot of changes in their agriculture. These are
changes at political, economic and technical level. This
means that information systems on agriculture (databases,
models, etc.) have to move along with those changes.
Consequently, the economic models in Estonia have to be
created, developed and renewed, and must be harmonised
with the European requirements. Hopefully, new infor
mation technology can be used to lead such evolutions.
A variation in the behavioural characteristics of agri
cultural production systems over time as well as between
countries is recognised. The diverse nature of agricultural
production systems and agrifood markets across the EU
poses a challenge to anyone seeking to develop a model
that can be used to analyse policy at the EU and its mem
ber state level.
The guiding principle in constructing the national
level commodity models is that the models are first and
foremost economic models and such economic theory is
our first guide in specifying the models. Economic rela
tionships in the national commodity models are based, in
so far as is practicable, on time series econometric esti
mates of these relationships. Theory and expert judgement
are also used in the verification and, if necessary, adjust
ment of econometrically estimated equations, particularly
when used to generate projections for further periods.
Improving the competitiveness of Estonian agriculture
is a priority objective of Estonian agricultural policy. The
outcome and impacts of those policy actions will strongly
depend on development of agricultural world markets.
Dairy sector is the most competitive branch of Esto
nian agriculture. Consequently, the need to make Esto
nian dairy farms more competitive is obvious.
New data analysis procedures provided by current data
mining (DM) (Andriaans and Zantinge 2003, Dunham
2003, Fayyad et al. 1996, Friedman 1997) have substan
tially changed the situation in the field of data processing
(DP). The situation in data mining is the most challenging
one. Data mining, often called knowledge discovery in
databases (KDD), started to depart from the statistics and
machine learning ghettos and move into the mainstream
almost 10 years ago.
Data mining is the process of discovery of useful in
formation from large collections of data. It has common
frontiers with several fields including Data Base Man
agement (DBM), Artificial Intelligence (AI), Machine
Learning (ML), Pattern Recognition (PR), and Data
Visualisation (DV).
The researchers of the Institute of Informatics of the
EAU have investigated the possibilities of using some
new DM methods and also have some experience in im
plementing algorithms used in DM packages (Bayesian
statistical methods (Põldaru and Roots 2001b, 2001c,
2003b), neural networks (Põldaru and Roots 2002a,
2003a), principal components method (Põldaru and Roots
2001a), decision trees and rules (e.g. CART – classification
Economics and rural development Vol. 1, No 1, 2005 ISSN 18223346
24
and regression trees) (Põldaru et al. 2003a, 2003d), asso
ciation rules discovery (Põldaru et al. 2003b), fuzzy re
gression methods (Põldaru et al. 2004b) and support vec
tor machine regression (Põldaru et al. 2004a, 2004b)). At
Estonian Agricultural University, some experience has
been gained in teaching the new data analysis procedures
(principal component analysis, Bayesian methods, neural
networks) for postgraduate students (Põldaru and Roots
2001b, 2003b, Põldaru et al. 2003c, Põldaru et al. 2004a).
The results of the research are published in many papers
and conference theses (see references by Põldaru, Roots,
Ruus and Viira).
Considering the reasons outlined above, there is a con
tinuing need for the application of more accurate and in
formative estimation techniques to econometric analysis.
The objective of this study is estimation of parameters
of econometric model of average total milk cost and analysis
of the results. The paper provides an overview about the
support vector machines regression (SVMR), describes the
potential implementation of it in rural areas and discusses
the implementation of this method for analysing the dairy
sector in Estonia (estimating econometric model of average
total milk cost). The data used is an unbalanced panel of
milk producers drawn from the FADN (Farm Accountancy
Data Network) database of Estonian milk producers. The pa
rameters are estimated on the basis of alternative models of
SVM regression using nonlinear model. The results are
compared mutually and with results of ordinary linear re
gression. For model (parameter) estimation the SVM mod
ule of Programming Environment R is used. R is an inte
grated suite of software facilities for data manipulation, cal
culation and graphical display.
Methods of investigation
Support vector machines have been successfully ap
plied to a number of applications ranging from particle
identification, face identification and text categorisation
to engine knock detection, bioinformatics and database
marketing. The approach is systematic and properly mo
tivated by statistical learning theory (Vapnik 1998).
Training (model parameter estimation) involves optimisa
tion of a convex cost function: there is no false local
minimum to complicate the estimation process. The ap
proach has many other benefits, for example, the model
constructed has an explicit dependence on the most in
formative patterns in the data (the support vectors), hence
interpretation is straightforward and data cleaning could
be implemented to improve performance. SVMs are the
best known from the class of algorithms, which use the
idea of kernel substitution and which we will broadly re
fer to as kernel methods.
Suppose beeing given statistical data {(x
1
, y
1
), . . . , (x
n
,
y
n
)}. The goal in SVM regression is to find the function
f(x) that has at most ε deviation from the actually obtained
targets y
i
for all the (training) data, and at the same time, is
as flat as possible. SVM regression uses the ε insensitive
loss function shown in Figure 1. If the deviation between
the actual and predicted value is less than ε , then the re
gression function is not considered to be in error.
Figure 1. A piecewise linear ε insensitive loss function and plot of
(
)
bxaxf
+
⋅
=
versus y with ε insensitive tube. Points out
side tube are errors
Thus, mathematically it looks like
εybxaε
ii
≤
−
+
⋅≤−
.
Geometrically, this can be visualized as a band or tube of
size 2ε around the hypothesis function f(x) and any points
outside this tube can be viewed as errors (Figure 1). All
training (data) points (x
i
, y
i
) for which
( )
εyxf
ii
≥−
are
known as support vectors; it is only these points that de
termine the parameters of f(x). In other words, errors are
not considered as long as they are less than ε, but any de
viation larger than this will not be accepted. To begin the
case of simple linear functions f(x), taking the following
form is described:
(
)
bxaxf +
⋅
=
(1).
For estimating the parameters of model (1) this prob
lem can be written as a convex optimization problem
(Vapnik 1998):
minimise
( )
∑
=
+⋅+
n
i
ii
ξξCa
1
*
2
0
Loss

ε
ε
=
0
5
10
15
0 5 10 15
ε
ε
Economics and rural development Vol. 1, No 1, 2005 ISSN 18223346
25
subject to
⎪
⎩
⎪
⎨
⎧
≥
+≤−+⋅
+≤−⋅−
0,
*
*
ii
iii
iii
ξξ
ξεybxa
ξεbxay
(2),
where ξ
i
; ξ
i
*
are slack variables to cope with otherwise in
feasible constraints of the optimization problem (2). The
constant C > 0 is specified beforehand. C is a regulariza
tion parameter that controls the tradeoff between the
flatness of f(x) and minimizing the training error. If C is
too small then insufficient stress will be placed on fitting
the training data. If C is too large then the algorithm will
overfit the training data. The formulation above corre
sponds to dealing with a so called εinsensitive loss func
tion
ε
ξ
described by
⎩
⎨
⎧
−
≤
=
otherwiseεξ
εξif
ξ
ε
....
........0
(3).
It turns out that the optimization problem (3) can be
solved more easily in its dual formulation. Hence a stan
dard dual method utilizing Lagrange multipliers will be
used. In the case of the Lagrangian dual (supporting) op
timisation problem needs to be optimised:
maximise
( ) ( )
( ) ( )
∑ ∑
∑∑
= =
= =
−⋅++−
−
⎪
⎩
⎪
⎨
⎧
⋅⋅−⋅−−
n
i
n
i
iiii
n
i
n
j
jijjii
ααyααε
xxαααα
1 1
**
1 1
**
2
1
subject to
( )
⎪
⎩
⎪
⎨
⎧
≤≤
=−
∑
=
Cαα
αα
ii
n
i
ii
*
1
*
,0
0
(4),
where α
i
and α
i
*
are Lagrangian multiplier.
The value of regression parameter a and predicted
value f(x) can be calculated as follows:
( )
∑
=
⋅−=
n
i
iii
xααa
1
*
(5)
and
( )
( )
∑
=
+⋅⋅−=
n
i
iii
bxxααxf
1
*
(6).
This is the socalled support vector expansion, i.e.
the regression coefficient a can be completely described
as a linear combination of the training patterns x
i
. In a
sense, the complexity of a function's representation by
SVs is independent of the dimensionality of the input
space X, and depends only on the number of SVs. More
over, the complete algorithm can be described in terms of
dot products between the data. Even when evaluating f(x)
explicit computing of a is not needed (although this may
be computationally more efficient in the linear setting).
These observations will come handy for the formulation
of a nonlinear extension.
The next step is to make the SVM algorithm non
linear. This, for instance, could be achieved by simply
preprocessing the training patterns x
i
by a map into some
feature space F, as described in (Vapnik 1998) and then
applying the standard SVM regression algorithm.
Firstly, a mapping must be defined from the space X
of regressors to the possibly infinite dimensional hy
pothesis space H, in which an inner product < , > is de
fined. This map is formally described as
Η
Χ
→:Φ
or
( )
xΦx
a
(7).
The choice of regression function f(x) is limited to
the class of functions which can be expressed as inner
products in H, taken between some weight vector a and
the mapped regressor
(
)
xΦ
:
(
)
(
)
bxΦaxf +=
,
(8).
The regression function in the hypothesis space is
consequently linear, and thus the nonlinear regression
problem of estimating
(
)
xf
has become a linear regres
sion problem in the hypothesis space H. Note that the
mapping
(
)
⋅
Φ
need never be computed explicitly; in
stead, the fact that if H is the reproducing kernel Hilbert
space induced by
(
)
⋅
⋅
,k
, then writing
( )
(
)
⋅
=
,
xkxΦ
is
used. This gives
(
)
(
)
(
)
jiji
xxkxΦxΦ
,,
=
(9).
The latter requirement is met for kernels fulfilling the
Mercer conditions (Vapnik 1998). These conditions are
satisfied for a wide range of kernels, including Gaussian
radial basis function (RBF)
(
) ( )
{
}
2
exp,
jiji
xxγxxk −⋅−=
(10),
and polynomial function
(
) ( )
d
jiji
gxxγxxk +⋅⋅−=,
(11).
It is emphasised that the feature space need never be
defined explicitly, since only the kernel is used in SVM
regression algorithms. Indeed, it is possible for multiple
feature spaces to be included by a single kernel.
Consequently, this allows to rewrite the SV algo
rithm (formulas (4)…(6)) as follows:
Economics and rural development Vol. 1, No 1, 2005 ISSN 18223346
26
maximise
( ) ( )
( )
( ) ( )
∑∑
∑∑
==
= =
−⋅++⋅−
−
⎪
⎩
⎪
⎨
⎧
⋅−⋅−−
n
i
iii
n
i
ii
n
i
n
j
jijjii
ααyααε
xxkαααα
1
*
1
*
1 1
**
,
2
1
subject to
( )
⎪
⎩
⎪
⎨
⎧
≤≤
=−
∑
=
Cαα
αα
ii
n
i
ii
*
1
*
,0
0
(12),
and regression parameter a and predicted value
(
)
xf
can
be calculated as follows:
( )
( )
∑
=
⋅−=
n
i
iii
xΦααa
1
*
(13)
( )
( )
( )
∑
=
+⋅−=
n
i
iii
bxxkααxf
1
*
,
(14).
The difference to the linear case is that a is no longer
explicitly given. However due to the theorem of
FischerRiesz (see e.g. (Riesz and Nagy, 1955)) it is al
ready uniquely defined in the weak sense by the dot
products
( )
xΦa,
. Also note that in the nonlinear set
ting, the optimization problem corresponds to finding the
flattest function in feature space, not in input space.
Results and discussion
Next the potential implementation of SVM regres
sion in rural areas is considered and the implementation
of this method for estimating an econometric model of
the average total milk cost (average total milk cost per kg
output) is discussed.
The econometric model is defined by
77665545
3322110
xbxbxbxb
xbxbxbby
⋅+⋅+⋅+⋅+
+⋅+⋅+⋅+=
(15),
where
y represents the average total milk cost per unit of
output (Estonian kroons per kg output),
x
1
represents the average milk yield per cow (kg),
x
2
represents the farm total labor input (hours per
hectare),
x
3
represents the manufactured (purchased) milk
price (Estonian kroons per 100 kg milk),
x
4
represents the total labor input per 100 kg of milk
(hours),
x
5
represents the wage per hour (Estonian kroons),
x
6
represents the total costs of feed per cow (Esto
nian kroons),
x
7
represents the invested capital per hectare (Esto
nian kroons).
The data is an unbalanced panel of milk producers
drawn from the FADN (Farm Accountancy Data Net
work) database of Estonian milk producers. Some previ
ous studies (Põldaru et al. 2004b, Viira 2003) have also
based on the FADN database. The characteristics of the
data are reported in Table 1.
Table 1. Data summary statistics
Statiatic Y x
1
x
2
x
3
x
4
x
5
x
6
x
7
Mean 2.11 5128 127.7 264.7 5.1 10.4 6889.5 16184.2
Median 1.99 5013.8 87.2 274.6 4.7 8.1 6122 10189.1
Standard Deviation 0.68 1154.6 115.7 53.8 2.6 9.3 3486.6 17180.8
Min 0.81 2447.3 13.2 95.3 0.7 6.0 613.8 425.2
Max 3.99 9706.7 769.8 409.4 16 44.4 22243.7 115657.9
The total number of observations n = 436.
In the case of the average total milk cost the nonlinear
functions are the most acceptable and must exhibit charac
teristics stated in the law of diminishing returns. According
to the law of diminishing returns when one or more vari
able inputs are added to one or more fixed inputs the extra
production obtained will, after a point, decline.
For linear and nonlinear model (parameter) estima
tion the SVM module (Meyer 2003) of Programming En
vironment R (Venables et al. 2003) is used. R is an inte
grated suite of software facilities for data manipulation,
calculation and graphical display.
When using SVM module for any given task, it is
always necessary to specify a set of parameters (the pa
rameters must be chosen in advance). Normally the archi
tecture of the SVM is specified in advance and weights
and biases are estimated by supervised learning. These
parameters include such indexes as whether one is inter
ested in regression estimation or pattern recognition, what
kernel is used, what scaling is to be done on the data, etc.
Previous studies (Põldaru et al.. 2004a, 2004b) show
that the most influential parameters are kernel type, pa
rameter gamma and parameter epsilon. The some non
linear SVM models are sensitive to “overfitting” and
Economics and rural development Vol. 1, No 1, 2005 ISSN 18223346
27
polynomial kernel function gave more acceptable results.
Previous experience has been considered for selecting pa
rameter sets in Table 2.
The summary of (given) parameter set for various vari
ants of the nonlinear model is reported in Table 2. Two
other parameters (parameter g and d in formula (11)) have
constant values for all alternatives, whereas g = 5, and d = 3.
The SVM models in Table 2 are compared with the
results of neural network models (varnn1 and varnn2) and
ordinary linear regression (OLS). In the case of alterna
tive “varnn1” there is one hidden node and in the case of
alternative “varnn2” – two hidden nodes.
The study shows that the parameter sets in Table 2
give the most acceptable results.
Table 2 presents also result summaries of the results
of various model alternatives. Summary characteristics
for various alternatives are number of support vectors and
coefficient of determination R
2
.
Table 2. Parameter set for various alternatives and summary characteristics of the models
Specified set of parameters Summary characteristics
Variant
Kernel type Epsilon Gamma Cost C Number of SV R
2
var1 Polynomial 0.5 0.08 1 61 0.886
var2 Polynomial 0.4 0.08 1 79 0.900
var3 Polynomial 0.3 0.08 1 122 0.909
var4 Radial 0.3 0.40 1 184 0.921
var5 Linear 0.3 0.14 1 188 0.806
varnn1 x x x x x 0.818
varnn2 x x x x x 0.848
OLS x x x x x 0.804
* parameters epsilon, gamma and C are specified for standardised data
Further the summary characteristics in Table 2 are
discussed. For different alternatives the number of sup
port vectors are different. The number of support vector
depends mainly on the value of parameter epsilon (ε).
The SVM models have, in general, offered greater accu
racy than have their statistical forebears (OLS and neural
network models). The values of the coefficient of deter
mination R
2
are higher than in linear model case (OLS
and var4 in Table 2). The minimal value in Table 2
(0.886) for nonlinear SVM models is higher than maxi
mal value for linear models (0.806). Consequently, the
nonlinear SVM regression models work well.
For econometric models the values of the coefficient
of determination R
2
are not the only characteristic for esti
mating the models generalizing capacity. Next exhibiting
of the characteristics, stated in the law of diminishing re
turns, by selected models is analysed. That may be done by
calculating the rules for partial derivatives of the models.
Next the partial derivatives for average milk yield per
cow (x
1
) for considered alternatives are computed and ana
lyzed. The values of partial derivatives are computed from
predicted values numerically. The graph of derivatives
with respect to independent variables for average milk
yield per cow (x
1
) is shown in Figure 2. The dot line on the
graphs presents OLS regression coefficient.
From the economic point of view the derivative for
average milk yield per cow, as for a production factor or
resource, must be negative (increasing the milk yield the
milk cost decreases) and increase (increasing the milk
yield the cost decrease diminish).
Figure 2. Graphs of partial derivatives with respect to value of independent variable for average milk yield per cow
1,2
1
0,8
0,6
0,4
0,2
0
0,2
0,4
3 2 1 0 1 2 3
Avera
g
e milk
y
ield per cow
Partial derivatives for average
milk yield per cow
Var3
Varnn2
Varnn1
OLS
Var4
Economics and rural development Vol. 1, No 1, 2005 ISSN 18223346
28
From Figure 2, it can be seen that when the Gaussian
radial basis kernel is used (var4), the graph of the partial
derivative is essentially nonlinear (can not be founded
from economic point of view), differ substantially from
other graphs on Figure 2 and varies within [–0.73, 0.33]
(has positive values). Consequently, the case of “overfit
ting” is observed. Although, in this case, (of the al
ternative var4 ) the value of the coefficient of determina
tion is maximal (0.921) that alternative is not acceptable
from economic point of view. Therefore, for estimating
the parameters of econometric model the Gaussian radial
basis kernel can not be recommended.
From Figure 2, it can be seen that when the neural
network models are used (varnn1 and varnn2), values of
derivatives are negative and the graphs of the partial de
rivative are concave (can not be founded from economic
point of view).
From the economic point of view the alternative var3
is the most acceptable one.
The study shows that the graphs of derivatives with
respect to the other independent variables behave analo
gously. Consequently, the most acceptable kernel from
economic point of view is polynomial kernel, and in the
following discussion the potential implementation of al
ternative var1, var2 and var3 (based on polynomial ker
nel) is considered.
Next the partial derivative for the most essential in
dependent variables is computed and analyzed: average
milk yield per cow (x
1
), total labor input per 100 kg of
milk (x
4
), the wage per hour (x
5
), and the total costs of
feed per cow (x
6
) for considered alternatives.
The graph of derivatives with respect to independent
variables for average milk yield per cow is shown in Figure 3.
Figure 3. Graphs of partial derivatives with respect to value of independent variable for average milk per cow
Analysis of the graphs may bring to the following
conclusions:
• The graphs of partial derivatives are analogous
(moderately nonlinear and convex).
• The nonlinear SVM regression models with lower
value of epsilon are more flexible (the value of de
rivative varies more (see var3).
• From the economic point of view the values of partial
derivative for average milk yield per cow, as for a
production factor, are negative (increasing the milk
yield the milk cost decreases) and increase (increas
ing the milk yield the cost decrease diminishes).
Consequently, every relation is acceptable and the
considered alternatives can be recommended for
practical use.
Next the partial derivative with respect to independ
ent variable for total labor input per 100 kg of milk (x
4
)
is computed and analyzed.
The graph of the partial derivative with respect to the
independent variables for total labor input per 100 kg of
milk is shown in Figure 4.
The graph shows that the relation for alternative var2
and var3 is essentially nonlinear. At the same time for al
ternatives var1 graph is moderately nonlinear and de
creasing (Figure 4). Consequently, all considered alterna
tives are acceptable from economic point of view.
Figure 5 shows the graphs of partial derivatives with
respect to wage per hour (x
5
).
The last graph shows that all relations are moderately
nonlinear and convex. The graphs of partial derivatives
for alternative var2 and var3 have a tendency to increase.
At the same time in the case of alternative var1 the graph
is decreasing and, consequently, that variant is acceptable
from economic point of view.
The graph of derivatives with respect to independent
variables for total costs of feed per cow (x
6
) is shown in
Figure 6.
1
0,8
0,6
0,4
0,2
0
3 2 1 0 1 2 3
Milk yield per cow
Partial derivatives for
Milk yield per cow
Var1
Var2
Var3
OLS
Economics and rural development Vol. 1, No 1, 2005 ISSN 18223346
29
Figure 4. Graphs of partial derivatives with respect to value of independent variable for total labor inputs (hours per 100 kg milk pro
duced)
Figure 5. Graphs of partial derivatives with respect to value of independent variable for wage per hour
(Estonian kroons per hour)
Figure 6. Graphs of partial derivatives with respect to value of independent variable for total cost of feed per cow
(Estonian kroons)
Analysis of these graphs may bring to the following
conclusions:
• The graphs of partial derivatives are analogous and
moderately nonlinear.
0
0,1
0,2
0,3
0,4
0,5
3 2 1 0 1 2 3
Total labor inputs (hours per 100 kg of milk)
Partial derivatives for total
labor inputs
Var3
Var1
Var2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
3 2 1 0 1 2 3
Wage per hour
Partial derivatives for
wage per hour
Var2
Var3
Var1
OLS
0,7
0,8
0,9
1
1,1
1,2
3 2 1 0 1 2 3
Total cost of feed per cow
Partial derivatives for
total cost of feed per cow
OLS
Var1
Var3
Var2
Economics and rural development Vol. 1, No 1, 2005 ISSN 18223346
30
• From the economic point of view the values of par
tial derivative for total costs of feed per cow, as for a
production factor, are positive (increasing the total
costs of feed per cow, the milk cost increases) and
decrease (increasing the total costs of feed per cow
the cost decrease diminish). Consequently, every re
lation is acceptable and the considered alternatives
can be recommended for practical use.
Figures 3, 4, 5 and 6 and Table 2 show that the con
sidered models are acceptable from economic point of
view and the models have high generalising capacity (R
2
varies within [0.886, 0.909]. In comparison with the other
methods, the SVM regression gives better results.
From considered alternative models the most suitable
for practical use are the alternatives var1 and var2 when
there is polynomial kernel, epsilon = 0.5 (va1) and epsi
lon = 0.4 (var2) and gamma = 0.08.
Conclusions
SVM regression provides a new approach to the
problem of parameter estimation of linear and especially
nonlinear econometric models. In this paper a brief ex
position of SVM regression and their flexibility in han
dling economic data is given. Different SVM regression
models are used for estimation of the econometric model
of average total milk cost in Estonian farms. The results
are compared mutually and with results of ordinary linear
regression. The discussion may be summarised in the fol
lowing conclusions:
1. Application of the SVM classification and re
gression in many fields of science and engineering
(including econometrics) is rapidly increasing.
2. The SVM regression models may be used for es
timating the parameters of linear and nonlinear
econometric models.
3. The SVM regression estimates and the least
square estimates of econometric model of average to
tal milk cost are similar, whereas the estimates for
some independent variables are essentially equiva
lent.
4. Using of polynomial kernel function gives more
acceptable results.
5. The nonlinear SVM regression models are sen
sitive to “overfitting”.
6. The suitable parameter selection allows diminish
the “overfitting” problem.
7. Programming Environment R can be used to
find the model parameter values.
8. Model combination and Bayesian methods can
partially overcome these methods, but require many
models to be trained and are hence computationally
expensive.
9. SVM regression, as a potential model estimation
method, can replace neural networks to solve non
linear problems in econometric modelling.
This analysis has demonstrated that interesting new
methods can be implemented for parameter estimation of
econometric models. This paper is expected to encourage
the use of SVM regression for econometric analysis.
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